Practice Exponential Function in Math

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

Quick Recap

An exponential function has the form f(x)=aโ‹…bxf(x) = a \cdot b^x where b>0b > 0, bโ‰ 1b \neq 1. The variable is in the exponent, not the base.

Growth (or decay) that multiplies by a constant factor repeatedly.

Showing a random 20 of 50 problems.

Example 1

challenge
Two species of bacteria start with the same population. Species A doubles every 44 hours, Species B triples every 99 hours. After how many hours do their populations match again, other than at t=0t=0? Express the smallest positive answer.

Example 2

hard
A bacterial culture grows from 500500 to 32,00032{,}000 in 55 hours. Assuming exponential growth, find the doubling time.

Example 3

easy
What is the initial value aa in f(x)=7โ‹…2xf(x) = 7 \cdot 2^x?

Example 4

medium
What is the range of f(x)=2xf(x) = 2^x?

Example 5

easy
Is f(x)=2xf(x)=2^x or g(x)=x2g(x)=x^2 the exponential function?

Example 6

medium
The population of a town grows from 50005000 to 66556655 in 33 years exponentially. Find the annual growth rate.

Example 7

easy
Evaluate 252^5.

Example 8

hard
For what value of bb does f(x)=bxf(x) = b^x pass through (3,64)(3, 64)?

Example 9

easy
A bacteria population starts at 500 and doubles every 3 hours. Write an exponential model and find the population after 9 hours.

Example 10

medium
Solve 32xโˆ’1=813^{2x - 1} = 81.

Example 11

challenge
If ax=3a^x=3 and ay=5a^y=5, find a2x+ya^{2x+y}.

Example 12

medium
A quantity halves every 55 years from 8080 grams. How much remains after 1515 years?

Example 13

easy
What is the domain of f(x)=2xf(x) = 2^x?

Example 14

medium
Solve 5x+1=1255^{x+1} = 125.

Example 15

medium
Simplify (23)2(2^3)^2.

Example 16

challenge
Solve 9x=3x+19^x=3^{x+1}.

Example 17

medium
A radioactive substance has a half-life of 10 years. If the initial amount is 200 g, how much remains after 30 years?

Example 18

medium
Identify the growth factor of g(x)=10โ‹…(1.25)xg(x) = 10 \cdot (1.25)^x.

Example 19

medium
Solve 22xโˆ’5โ‹…2x+4=02^{2x}-5\cdot 2^x+4=0.

Example 20

medium
Solve 4x=84^x = 8.