Math: Logarithms and Exponential Equations
Using logarithms to rewrite, evaluate, and solve exponential relationships
Math: Logarithms and Exponential Equations
Using logarithms to rewrite, evaluate, and solve exponential relationships
Math - Grade 9-12
- 1
Rewrite the exponential equation 2^5 = 32 in logarithmic form.
Use the pattern a^b = c becomes log_a(c) = b.
The logarithmic form is log_2(32) = 5 because 2 raised to the power of 5 equals 32. - 2
Rewrite the logarithmic equation log_3(81) = 4 in exponential form.
The exponential form is 3^4 = 81 because a logarithm gives the exponent needed to produce the number. - 3
Evaluate log_10(1000).
Ask yourself what power of 10 gives 1000.
The value of log_10(1000) is 3 because 10^3 = 1000. - 4
Evaluate log_4(64).
The value of log_4(64) is 3 because 4^3 = 64. - 5
Solve the equation 5^x = 125.
Rewrite 125 as a power of 5.
The solution is x = 3 because 125 = 5^3. - 6
Solve the equation 2^(x + 1) = 16.
The solution is x = 3 because 16 = 2^4, so x + 1 = 4 and x = 3. - 7
Solve the equation 3^(2x) = 81.
Write 81 as a power of 3 first.
The solution is x = 2 because 81 = 3^4, so 2x = 4 and x = 2. - 8
Solve the equation 10^x = 7. Round your answer to the nearest thousandth.
The solution is x = log(7), which is approximately 0.845 to the nearest thousandth. - 9
Evaluate ln(e^6).
ln and e raised to a power undo each other.
The value of ln(e^6) is 6 because the natural logarithm and the exponential function with base e are inverse operations. - 10
Evaluate e^(ln 9).
The value of e^(ln 9) is 9 because the exponential function with base e and the natural logarithm are inverse operations. - 11
Use the product rule to expand log_2(8x).
Use log_b(MN) = log_b(M) + log_b(N).
The expanded form is log_2(8) + log_2(x), which simplifies to 3 + log_2(x) because log_2(8) = 3. - 12
Use the power rule to expand log_5(x^3).
The expanded form is 3 log_5(x) because the exponent becomes a coefficient. - 13
Condense the expression 2 ln(x) + ln(3) into a single logarithm.
First use the power rule, then use the product rule.
The condensed expression is ln(3x^2) because 2 ln(x) = ln(x^2) and ln(x^2) + ln(3) = ln(3x^2). - 14
Solve for x: log_2(x) = 6.
The solution is x = 64 because rewriting in exponential form gives 2^6 = x. - 15
Solve for x: ln(x) = 2.
Rewrite the equation in exponential form using base e.
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.