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≈2.71828e \approx 2.71828: ln⁡x=log⁡ex\ln x = \log_e x. It is the inverse function of exe^x.

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

Showing a random 20 of 50 problems.

Example 1

medium
Simplify eln⁡5+ln⁡2e^{\ln 5 + \ln 2}.

Example 2

easy
What is ln⁡(1/e)\ln(1/e)?

Example 3

easy
Evaluate ln⁡1\ln 1.

Example 4

hard
Carbon-14 has half-life 57305730 years. A sample contains 30%30\% of its original C-14. Find its age using N=N0e−ktN = N_0 e^{-kt}.

Example 5

medium
Is ln⁡(2)+ln⁡(5)=ln⁡(10)\ln(2) + \ln(5) = \ln(10)?

Example 6

hard
Solve ln⁡(x)+ln⁡(x−3)=ln⁡(2x+8)\ln(x) + \ln(x - 3) = \ln(2x + 8) for xx.

Example 7

challenge
Solve e2x−5ex+6=0e^{2x} - 5e^x + 6 = 0 for all real xx.

Example 8

medium
Simplify ln⁡(e3x2)−ln⁡x\ln(e^3 x^2) - \ln x.

Example 9

medium
If ln⁡2≈0.693\ln 2 \approx 0.693, estimate ln⁡8\ln 8.

Example 10

easy
Rewrite ln⁡x\ln\sqrt{x} using a logarithm property.

Example 11

medium
Find the domain of f(x)=ln⁡(x2−4)f(x) = \ln(x^2 - 4).

Example 12

medium
Solve ln⁡(2x+1)=3\ln(2x + 1) = 3 for xx.

Example 13

medium
Solve ln⁡x=2\ln x = 2 for xx.

Example 14

hard
What is the slope of the tangent to y=ln⁡xy = \ln x at x=2x = 2?

Example 15

easy
Expand ln⁡(2x3)\ln(2x^3).

Example 16

medium
Solve e2x=7e^{2x} = 7 for xx.

Example 17

hard
Use ln⁡\ln to solve 5x=3x+25^x = 3^{x+2}.

Example 18

medium
Solve ln⁡(x)+ln⁡(x−3)=ln⁡10\ln(x) + \ln(x - 3) = \ln 10 for xx.

Example 19

easy
Evaluate ln⁡(e5)\ln(e^5).

Example 20

easy
Evaluate ln⁡e\ln e.
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