Practice Natural Logarithm in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick Recap

The logarithm with base $e \approx 2.71828$ : $\ln x = \log_e x$ . It is the inverse function of $e^x$ .

If $e^x$ asks 'what do I get after growing continuously for time $x$ ?', then $\ln x$ asks 'how long do I need to grow continuously to reach $x$ ?' The natural log measures time in the world of continuous growth.

Showing a random 20 of 50 problems.

Example 1

medium

Simplify

e^{\ln 5 + \ln 2}

Example 2

easy

What is

\ln(1/e)

Example 3

easy

Evaluate

\ln 1

Example 4

hard

Carbon-14 has half-life

5730

years. A sample contains

30\%

of its original C-14. Find its age using

N = N_0 e^{-kt}

Example 5

medium

\ln(2) + \ln(5) = \ln(10)

Example 6

hard

Solve

\ln(x) + \ln(x - 3) = \ln(2x + 8)

for

x

Example 7

challenge

Solve

e^{2x} - 5e^x + 6 = 0

for all real

x

Example 8

medium

Simplify

\ln(e^3 x^2) - \ln x

Example 9

medium

\ln 2 \approx 0.693

, estimate

\ln 8

Example 10

easy

Rewrite

\ln\sqrt{x}

using a logarithm property.

Example 11

medium

Find the domain of

f(x) = \ln(x^2 - 4)

Example 12

medium

Solve

\ln(2x + 1) = 3

for

x

Example 13

medium

Solve

\ln x = 2

for

x

Example 14

hard

What is the slope of the tangent to

y = \ln x

x = 2

Example 15

easy

Expand

\ln(2x^3)

Example 16

medium

Solve

e^{2x} = 7

for

x

Example 17

hard

Use

\ln

to solve

5^x = 3^{x+2}

Example 18

medium

Solve

\ln(x) + \ln(x - 3) = \ln 10

for

x

Example 19

easy

Evaluate

\ln(e^5)

Example 20

easy

Evaluate

\ln e

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Related Concepts

Logarithm Euler's Number Change of Base Formula Solving Exponential Equations