Math: Logarithms and Exponential Equations Worksheet

Question 1

Rewrite the exponential equation 2^5 = 32 in logarithmic form.

Accepted Answer

The logarithmic form is log_2(32) = 5 because 2 raised to the power of 5 equals 32.

Question 2

Rewrite the logarithmic equation log_3(81) = 4 in exponential form.

Accepted Answer

The exponential form is 3^4 = 81 because a logarithm gives the exponent needed to produce the number.

Question 3

Evaluate log_10(1000).

Accepted Answer

The value of log_10(1000) is 3 because 10^3 = 1000.

Question 4

Evaluate log_4(64).

Accepted Answer

The value of log_4(64) is 3 because 4^3 = 64.

Question 5

Solve the equation 5^x = 125.

Accepted Answer

The solution is x = 3 because 125 = 5^3.

Question 6

Solve the equation 2^(x + 1) = 16.

Accepted Answer

The solution is x = 3 because 16 = 2^4, so x + 1 = 4 and x = 3.

Question 7

Solve the equation 3^(2x) = 81.

Accepted Answer

The solution is x = 2 because 81 = 3^4, so 2x = 4 and x = 2.

Question 8

Solve the equation 10^x = 7. Round your answer to the nearest thousandth.

Accepted Answer

The solution is x = log(7), which is approximately 0.845 to the nearest thousandth.

Question 9

Evaluate ln(e^6).

Accepted Answer

The value of ln(e^6) is 6 because the natural logarithm and the exponential function with base e are inverse operations.

Question 10

Evaluate e^(ln 9).

Accepted Answer

The value of e^(ln 9) is 9 because the exponential function with base e and the natural logarithm are inverse operations.

Question 11

Use the product rule to expand log_2(8x).

Accepted Answer

The expanded form is log_2(8) + log_2(x), which simplifies to 3 + log_2(x) because log_2(8) = 3.

Question 12

Use the power rule to expand log_5(x^3).

Accepted Answer

The expanded form is 3 log_5(x) because the exponent becomes a coefficient.

Question 13

Condense the expression 2 ln(x) + ln(3) into a single logarithm.

Accepted Answer

The condensed expression is ln(3x^2) because 2 ln(x) = ln(x^2) and ln(x^2) + ln(3) = ln(3x^2).

Question 14

Solve for x: log_2(x) = 6.

Accepted Answer

The solution is x = 64 because rewriting in exponential form gives 2^6 = x.

Question 15

Solve for x: ln(x) = 2.

Accepted Answer

The solution is x = e^2, which is approximately 7.389, because ln(x) = 2 means x is the number whose natural log is 2.

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