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A. Exponential and Logarithmic Functions · Exponential and Logarithmic Functions

Exponential and Logarithmic Functions: Unit Review

A1.1 A1.2 A1.3 A1.4 A2.1 A2.2 A2.3 A2.4 A3.1 A3.2 A3.3 A3.4About 40 minutes
Learning goals
  • I can evaluate logarithms and use the laws of logarithms.
  • I can graph logarithmic functions, their inverses and their transformations.
  • I can solve exponential and logarithmic equations and apply them to real problems.

Words to know

logarithm
log_b x = y means bʸ = x: a logarithm is an exponent.
inverse functions
y = bˣ and y = log_b x undo each other; their graphs are reflections in y = x.
extraneous solution
A value from the algebra that fails the original equation.

Let's review!

In this unit you learned that a logarithm is an exponent, evaluated logs by hand and with the change of base formula, and switched between exponential and logarithmic form. You proved and used the laws of logarithms, graphed y = log_b x as the inverse of y = bˣ, and transformed it with y = a log (k(x − d)) + c. Then you solved exponential equations (with a common base or by taking logs), solved logarithmic equations while rejecting extraneous roots, and applied it all to pH, decibels, earthquake magnitude, half-life, compound interest and cooling.

This review has three questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Work through one or two sections at a time with a scientific calculator. Give exact answers when you can (for example x = log 3 ÷ log 2) and round decimals only at the end, to three decimal places unless the context says otherwise. For every equation, check your answer by substituting it into the original equation. For each question you miss, write down why (wrong form? wrong law? forgot to check the domain?) and revisit that lesson. Finish with the 10-question quiz.

A1 Evaluatelogs are exponents (1)LOG key, change of base (2)switch forms (3)laws of logs (4, 5)A2 Graphy = logb x (6)inverse of bˣ (7)transformations (8, 9)log scales (14)A3 Solveequivalent forms (10)common base (11)take logs (12)log equations (13)growth, decay (15)numbers in brackets = lesson numbers
The unit in one picture: evaluating logarithms (A1) leads to graphing them (A2) and solving equations and problems with them (A3).
Just for fun

You can now find the age of ancient charcoal, how loud a concert really is, how much stronger one earthquake is than another, and when your hot chocolate will be drinkable. Not bad for a button marked LOG.

Logarithms: The Exponent You Are Looking For

1

Evaluate log₅ 125.

2

Evaluate log₂ (1/32).

3

Evaluate log₂₇ 3.

Common Logarithms and Logarithms to Any Base with Technology

1

Evaluate log 0.0045 to four decimal places.

2

Evaluate log₇ 100 to four decimal places.

3

Between which two whole numbers is log₂ 100?

Switching Between Exponential and Logarithmic Form

1

Write 4^(−1/2) = 1/2 in logarithmic form.

2

Solve log_x 64 = 3.

3

Solve 6ˣ = 20 to three decimal places.

The Laws of Logarithms

1

Evaluate log₄ 2 + log₄ 32.

2

Evaluate log₂ ∛8.

3

Disprove log (M/N) = log M ÷ log N with a counterexample.

Simplifying and Evaluating with the Laws of Logarithms

1

Evaluate log₃ 6 + log₃ 4.5.

2

Evaluate 2 log 5 + log 40 − log 10.

3

Given log 2 ≈ 0.3010 and log 3 ≈ 0.4771, find log 54.

The Graph of y = log_b x and Its Key Features

1

State the domain, range and asymptote of y = log₈ x.

2

The graph of y = log_b x passes through (1/9, −2). Find b.

3

Find the average rate of change of y = log₂ x from x = 8 to x = 16.

Exponential and Logarithmic Functions as Inverses

1

Find the inverse of f(x) = 3ˣ + 1.

2

Find the inverse of g(x) = log₂ (x + 4).

3

The point (2, 9) is on y = 3ˣ. Which point is on y = log₃ x?

xy−24681012−24681012(0, 1)(1, 10)(1, 0)(10, 1)y = 10ˣy = log xy = x
The two functions at the heart of the unit: y = 10ˣ and its inverse y = log x, mirror images in the line y = x. (0, 1) on one matches (1, 0) on the other, and (1, 10) matches (10, 1).

Transforming Logarithmic Graphs I: Translations, Stretches and Reflections

1

Find the images of (0.1, −1), (1, 0) and (10, 1) under y = −log (x + 1) + 2.

2

State the asymptote and domain of y = −log (x + 1) + 2.

3

Find the x-intercept of y = log (x − 5) + 1.

Transforming Logarithmic Graphs II: Horizontal Stretches, Reflections and All Four Parameters

1

Write y = log (3x + 12) in the form y = log (k(x − d)) and describe the transformations.

2

Find the image of (10, 1) under y = 2 log (−(x − 1)).

3

State the domain of y = 2 log (−(x − 1)).

Equivalent Expressions with Logarithms and Exponents

1

Expand log₂ (4x³/y) for x, y > 0.

2

Condense 3 log a − (1/2) log b.

3

For which x are log (x⁶) and 6 log x equivalent?

Solving Exponential Equations with a Common Base

1

Solve 4^(x − 1) = 32.

2

Solve 27ˣ = 9^(x + 2).

3

Solve 2^(x + 2) − 2ˣ = 24.

Solving Exponential Equations with Logarithms

1

Solve 5ˣ = 3^(x + 1) to three decimal places.

2

Solve 3(1.2)ˣ = 18 to three decimal places.

3

Solve 4ˣ − 3(2ˣ) − 4 = 0.

Solving Logarithmic Equations

1

Solve log₄ (x − 2) + log₄ (x + 4) = 2.

2

Solve log (x + 9) − log x = 1.

3

Solve log₂ (x² − 6x) = 4.

Logarithmic Scales: pH, Decibels and Earthquake Magnitude

1

Find the pH when [H⁺] = 2.5 × 10⁻⁸ mol/L.

2

How many times more intense is a 95 dB sound than a 65 dB sound?

3

Compare the amplitudes of earthquakes of magnitude 6.8 and 4.3.

Solving Growth and Decay Problems

1

A substance has a half-life of 5 days. How long until 10% remains?

2

$2500 is invested at 3.6% per year compounded monthly. How long until it doubles?

3

Tea at 85 °C cools in a 22 °C room; the difference halves every 10 minutes. When is it 30 °C?

Check your understanding

Quiz

1.

What is log₃ 81?

2.

Which expression equals log₆ 50?

3.

Evaluate log₂ 48 − log₂ 3.

4.

What is the vertical asymptote of y = 3 log (x − 2) + 5?

5.

What is the inverse of f(x) = 10^(x + 1)?

6.

Solve 8ˣ = 4^(x + 1).

7.

To three decimal places, the solution of 3^(x − 1) = 20 is

8.

Solving log x + log (x − 4) = log 12 gives x = 6 or x = −2. Which is correct?

9.

A sound increases from 50 dB to 80 dB. Its intensity is multiplied by

10.

A substance has a half-life of 10 years. How long until 1/16 of it remains?

For parents and teachers

Use the review to find what to revisit:

  • Can't evaluate simple logs like log₂ (1/8) → Lesson 1 (Logarithms: The Exponent You Are Looking For).
  • Wrong decimals with the calculator, or dividing the wrong way in change of base → Lesson 2 (Common Logarithms and Logarithms to Any Base).
  • Mixing up which number is the exponent when switching forms → Lesson 3 (Switching Between Exponential and Logarithmic Form).
  • Inventing a law for log (M + N), or misusing the power law → Lesson 4 (The Laws of Logarithms).
  • Combining logs before moving coefficients inside → Lesson 5 (Simplifying and Evaluating with the Laws of Logarithms).
  • Wrong domain, range or asymptote for y = log_b x → Lesson 6 (The Graph of y = log_b x).
  • Errors finding inverses, or asymptotes not swapped → Lesson 7 (Exponential and Logarithmic Functions as Inverses).
  • Shifting the wrong way, or the asymptote not moved → Lesson 8 (Transforming Logarithmic Graphs I).
  • Reading log (2x − 6) as a shift of 6, or multiplying by k in the mapping rule → Lesson 9 (Transforming Logarithmic Graphs II).
  • Splitting the log of a sum, or ignoring restrictions → Lesson 10 (Equivalent Expressions).
  • Missing a common base or a second solution → Lesson 11 (Exponential Equations with a Common Base).
  • Taking logs before isolating the power, or keeping 3ˣ = −1 → Lesson 12 (Exponential Equations with Logarithms).
  • Keeping extraneous roots → Lesson 13 (Solving Logarithmic Equations).
  • Treating pH, decibels or magnitude as linear → Lesson 14 (Logarithmic Scales).
  • Wrong model or wrong rate per period → Lesson 15 (Solving Growth and Decay Problems).

Looking ahead: logarithms come back in Calculus and Vectors (MCV4U), where you will meet the natural logarithm ln x and the derivatives of exponential functions. The next unit of this course, Radian Measure and Trigonometric Functions, starts the trigonometry strand.