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Exponential and logarithmic functions: mixed assessment

Read the idea, work independently, then explain what changed.

TOPIC 01

Exponential and logarithmic functions: mixed assessment

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

Connect the chapter skills without relying on the order of the exercises.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

Simplify the product.

2^{7}\cdot2^3

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use exponent laws only within their real-number domain.

Hint 2

Add exponents when multiplying equal bases.

Worked solution

Use exponent laws only within their real-number domain.
Calculate or simplify this relation.
$2^{7+3}=2^{10}=1024$
The bases must be the same for this rule.

The requested value is 1024.

Checks and common pitfalls: The bases must be the same for this rule.

Think first. Reveal a hint when the class is ready.

02 / Foundation#Your turn

Solve the exponential equation.

2^x=2^{9}

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Identify the base and whether the exponential graph grows or decays.

Hint 2

Equal outputs of a strictly monotone exponential have equal inputs.

Worked solution

Identify the base and whether the exponential graph grows or decays.
Calculate or simplify this relation.
$x=9$
The base is positive and not one.

The requested value is 9.

Checks and common pitfalls: The base is positive and not one.

Think first. Reveal a hint when the class is ready.

03 / Foundation#Your turn

Evaluate by change of base.

\log_{128}16384

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Rewrite logarithms as exponent equations and check base and argument conditions.

Hint 2

Use base two in numerator and denominator.

Worked solution

Rewrite logarithms as exponent equations and check base and argument conditions.
Calculate or simplify this relation.
$\frac{\log_216384}{\log_2128}=\frac{14}{7}=2$
A change of base needs the same new base in both places.

The requested value is 2.

Checks and common pitfalls: A change of base needs the same new base in both places.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

Solve the inequality.

\log_{1/2}(x-7)>1

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use logarithmic monotonicity after enforcing a positive argument.

Hint 2

The logarithm decreases, so the comparison reverses.

Worked solution

Use logarithmic monotonicity after enforcing a positive argument.
Calculate or simplify this relation.
$0<x-7<1/2$
The argument positivity supplies the lower bound.

7<x<7+1/2.

Checks and common pitfalls: The argument positivity supplies the lower bound.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Solve both parts and justify the conditions used. Part A: Show that f(x)=2ˣ−3 has a zero between 1 and 2, and explain what else makes the zero unique. Part B: A population is 800 and increases by 10% each year. Find it after two years.

\begin{gathered}\text{A: }f(x)=2^x-3\end{gathered}

Use the domain, units and sampling assumptions stated in the question.
Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Translate the context into an exponential or logarithmic equation and check what the model assumes.

Hint 2

B: Translate the context into an exponential or logarithmic equation and check what the model assumes.

Worked solution

Part A reasoning
Translate the context into an exponential or logarithmic equation and check what the model assumes.
Calculate or simplify this relation.
$f(1)=-1<0,\quad f(2)=1>0$
A sign change alone does not establish uniqueness.
Part B reasoning
Translate the context into an exponential or logarithmic equation and check what the model assumes.
Calculate or simplify this relation.
$800(1.1)^2=968$
This is compounding, not a single 20% increase.

A: Continuity gives existence; strict increase gives uniqueness. B: The requested value is 968.

Checks and common pitfalls: A sign change alone does not establish uniqueness. This is compounding, not a single 20% increase.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

Simplify for a>0.

\frac{a^{12}}{a^{8}}

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use exponent laws only within their real-number domain.

Hint 2

Subtract exponents when dividing equal nonzero bases.

Worked solution

Use exponent laws only within their real-number domain.
Calculate or simplify this relation.
$a^{12-8}=a^4$
The nonzero base is required for division.

a⁴.

Checks and common pitfalls: The nonzero base is required for division.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

07 / Standard#Your turn

Find the y-intercept.

y=8\cdot3^x

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Identify the base and whether the exponential graph grows or decays.

Hint 2

At a y-intercept, x=0.

Worked solution

Identify the base and whether the exponential graph grows or decays.
Calculate or simplify this relation.
$y(0)=8\cdot3^0=8$
The outside multiplier scales the graph vertically.

The requested value is 8.

Checks and common pitfalls: The outside multiplier scales the graph vertically.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

Evaluate the logarithm with a base below one.

\log_{1/8}64

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Rewrite logarithms as exponent equations and check base and argument conditions.

Hint 2

A negative exponent turns the small base into a large value.

Worked solution

Rewrite logarithms as exponent equations and check base and argument conditions.
Calculate or simplify this relation.
$(1/8)^{-2}=64$
Positive arguments can have negative logarithms.

The requested value is -2.

Checks and common pitfalls: Positive arguments can have negative logarithms.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

Solve both parts and justify the conditions used. Part A: Explain why the graphs y=log₂x and y=2ˣ are reflections in y=x. Part B: Find the inverse function and its domain.

\begin{gathered}\text{A: }y=\log_2x\iff x=2^y\\\text{B: }y=\log_9x\end{gathered}

Use the domain, units and sampling assumptions stated in the question.
Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Use logarithmic monotonicity after enforcing a positive argument.

Hint 2

B: Use logarithmic monotonicity after enforcing a positive argument.

Worked solution

Part A reasoning
Use logarithmic monotonicity after enforcing a positive argument.
Calculate or simplify this relation.
$(a,b)\mapsto(b,a)$
The axes and domains are exchanged, not simply translated.
Part B reasoning
Use logarithmic monotonicity after enforcing a positive argument.
Calculate or simplify this relation.
$x=9^y\Rightarrow f^{-1}(x)=9^x$
The original logarithm range becomes the inverse domain.

A: The input-output coordinates are exchanged by inversion. B: Inverse: y=9^x for all real x.

Checks and common pitfalls: The axes and domains are exchanged, not simply translated. The original logarithm range becomes the inverse domain.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

A population is 800 and increases by 10% each year. Find it after two years.

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate the context into an exponential or logarithmic equation and check what the model assumes.

Hint 2

The second percentage applies to the enlarged population.

Worked solution

Translate the context into an exponential or logarithmic equation and check what the model assumes.
Calculate or simplify this relation.
$800(1.1)^2=968$
This is compounding, not a single 20% increase.

The requested value is 968.

Checks and common pitfalls: This is compounding, not a single 20% increase.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

Compare the powers without decimal approximations.

(1/9)^2,\quad(1/9)^3

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use exponent laws only within their real-number domain.

Hint 2

The common base is between zero and one.

Worked solution

Use exponent laws only within their real-number domain.
Calculate or simplify this relation.
$(1/9)^2-(1/9)^3=(9-1)/729>0$
Increasing a positive exponent reduces a power with base between zero and one.

The squared value is larger.

Checks and common pitfalls: Increasing a positive exponent reduces a power with base between zero and one.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

12 / Standard#Your turn

Find the positive exponential base.

f(x)=a^x,\quad f(2)=81

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Identify the base and whether the exponential graph grows or decays.

Hint 2

A real exponential base is positive.

Worked solution

Identify the base and whether the exponential graph grows or decays.
Calculate or simplify this relation.
$a^2=81,\quad a>0\Rightarrow a=9$
The negative algebraic square root is not an admissible exponential base.

The requested value is 9.

Checks and common pitfalls: The negative algebraic square root is not an admissible exponential base.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

Simplify using the product rule.

\log_2512+\log_2 8

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Rewrite logarithms as exponent equations and check base and argument conditions.

Hint 2

Adding logarithms multiplies positive arguments.

Worked solution

Rewrite logarithms as exponent equations and check base and argument conditions.
Calculate or simplify this relation.
$\log_2(512\cdot8)=\log_24096=12$
It does not correspond to adding the arguments.

The requested value is 12.

Checks and common pitfalls: It does not correspond to adding the arguments.

Think first. Reveal a hint when the class is ready.

14 / Transfer#Your turn

Find the inverse function and its domain.

y=\log_9x

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use logarithmic monotonicity after enforcing a positive argument.

Hint 2

Swap input and output after solving for the original input.

Worked solution

Use logarithmic monotonicity after enforcing a positive argument.
Calculate or simplify this relation.
$x=9^y\Rightarrow f^{-1}(x)=9^x$
The original logarithm range becomes the inverse domain.

Inverse: y=9^x for all real x.

Checks and common pitfalls: The original logarithm range becomes the inverse domain.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

15 / Transfer#Your turn

Solve both parts and justify the conditions used. Part A: Find the base-10 logarithmic increase when a positive quantity is multiplied by 1000. Part B: Show that f(x)=2ˣ−3 has a zero between 1 and 2, and explain what else makes the zero unique.

\begin{gathered}\text{A: }\log_{10}(1000x)-\log_{10}x\\\text{B: }f(x)=2^x-3\end{gathered}

Use the domain, units and sampling assumptions stated in the question.
Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Translate the context into an exponential or logarithmic equation and check what the model assumes.

Hint 2

B: Translate the context into an exponential or logarithmic equation and check what the model assumes.

Worked solution

Part A reasoning
Translate the context into an exponential or logarithmic equation and check what the model assumes.
Calculate or simplify this relation.
$\log_{10}(1000x/x)=\log_{10}1000=3$
The result is independent of the initial positive x.
Part B reasoning
Translate the context into an exponential or logarithmic equation and check what the model assumes.
Calculate or simplify this relation.
$f(1)=-1<0,\quad f(2)=1>0$
A sign change alone does not establish uniqueness.

A: The requested value is 3. B: Continuity gives existence; strict increase gives uniqueness.

Checks and common pitfalls: The result is independent of the initial positive x. A sign change alone does not establish uniqueness.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

Think first. Reveal a hint when the class is ready.

End-of-lesson check and correction

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Choose a foundation skill to revisit ↗

Teacher preparation and assessment

Question sequence

Ask students to name the relevant condition before calculating.

Board plan

Compare valid methods and annotate their conditions.

Anticipated thinking

A correct final value may still hide a missing assumption.

Assessment checklist

Check the method, conditions, reasoning and interpretation separately.