HONORS ALGEBRA 2 : Solving Exponential Equations

Problems 1-4 : Solve the given exponential equation and round the answer to three decimal places. 

Problem 1 :

7 - 2e^x = 4

Solution :

7 - 2e^x = 4

Subtract 7 from both sides.

-2e^x = -3

Divide both sides by -3.

e^x = 1.5

We know that the base of a natural logarithm is e. Since the base on the right side of the equation above is e, we can take natural logarithm on both sides and solve the equation further.

ln e^x = ln 1.5

x ⋅ ln e = ln 1.5

x(1) = 0.405

x = 0.405

Problem 2 :

2^{3 - x} = 565

Solution :

2^{3 - x} = 565

Convert the above equation to exponential form.

3 - x = log₂ 565

Subtract 3 from both sides.

-x = log₂ 565 - 3

Multiply both sides by -1.

x = 3 - log₂ 565

x = -6.412

Problem 3 :

8^{log x} = 5

Solution :

8^{log x} = 5

Convert the above equation to logarithmic form.

log x = log₈ 5

log x = 0.773976

In the equation above, since the base of the logarithm is not mentioned, we have to understand that it is a common logarithm and base of a common logarithm is 10.

log₁₀ x = 0.773976

Convert the above equation to exponential form.

x = 10^0.773976

x = 5.943

Problem 4 :

10^3x = 1.5

Solution :

10^3x = 1.5

Convert the above equation to logarithmic form.

3x = log₁₀ 1.5

Divide both sides by 3.

x = 0.059

Problems 5-6 : Solve the given exponential equation and show work leading to exact answer.

Problem 5 :

Solution :

4^{3x - 1} = 4^-2

3x - 1 = -2

Add 1 to both sides.

3x = -1

Divide both sides by 3.

Problem 6 :

Solution :

3x² = 4x

Subtract 4x from both sides.

3x² - 4x = 0

x(3x - 4) = 0

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