Solve the following exponential equation. Express the solution set in terms of natural logarithms or common
logarithms. Then, use a calculator to obtain a decimal approximation for the solution

The solution set expressed in terms of logarithms is
(Use a comma to separate answers as needed. Simplify your answer. Use integers or decimals for any
numbers in the expression. Use In for natural logarithm and log for common logarithm.)

Ask by Lambert Bond. in the United States

Mar 31,2025

Question

Solve the following exponential equation. Express the solution set in terms of natural logarithms or common
logarithms. Then, use a calculator to obtain a decimal approximation for the solution

The solution set expressed in terms of logarithms is
(Use a comma to separate answers as needed. Simplify your answer. Use integers or decimals for any
numbers in the expression. Use In for natural logarithm and log for common logarithm.)

Ask by Lambert Bond. in the United States

Mar 31,2025

UpStudy AI · Accepted Answer

We start with the equation

$$
7^{0.9x}=411.
$$

Take the natural logarithm of both sides:

$$
\ln\left(7^{0.9x}\right)=\ln(411).
$$

Using the logarithm power rule $$\ln(a^b)=b\ln(a)$$, we have:

$$
0.9x\,\ln(7)=\ln(411).
$$

Solve for $$x$$ by dividing by $$0.9\ln(7)$$:

$$
x=\frac{\ln(411)}{0.9\,\ln(7)}.
$$

This is the solution expressed in terms of natural logarithms. Alternatively, using common logarithms:

$$
x=\frac{\log(411)}{0.9\,\log(7)}.
$$

Using a calculator, we find:

$$
\ln(411)\approx6.02,\quad \ln(7)\approx1.95.
$$

Thus,

$$
x\approx\frac{6.02}{0.9 \times 1.95}\approx\frac{6.02}{1.755}\approx3.43.
$$
The solution expressed in terms of natural logarithms is $$ x = \frac{\ln(411)}{0.9 \ln(7)} $$. Using a calculator, the decimal approximation is approximately 3.43.

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