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.
7^{2x} = 889
the solution set expressed in terms of logarithms is \boxed{}.
(use a comma to separate answers as needed. simplify your answer. use integers or fractions for any numbers in the equation. use ln for natural logarithm and log for common logarithm.)

Explanation:

Step1: Take natural log on both sides

To solve the exponential equation \(7^{2x}=889\), we start by taking the natural logarithm (ln) of both sides. This is a common technique for solving exponential equations because it allows us to use the property of logarithms \(\ln(a^b)=b\ln(a)\) to bring down the exponent.

Taking \(\ln\) of both sides gives us:

$$ \ln(7^{2x})=\ln(889) $$

Step2: Apply logarithm power rule

Using the power rule of logarithms \(\ln(a^b) = b\ln(a)\), we can simplify the left - hand side of the equation. For \(a = 7\) and \(b=2x\), we have:

$$ 2x\ln(7)=\ln(889) $$

Step3: Solve for x

Now, we want to isolate \(x\). We can do this by dividing both sides of the equation by \(2\ln(7)\).

$$ x=\frac{\ln(889)}{2\ln(7)} $$

If we want to get a decimal approximation, we can use a calculator to find the values of \(\ln(889)\) and \(\ln(7)\).

We know that \(\ln(889)\approx6.785\) and \(\ln(7)\approx1.9459\)

$$ x=\frac{6.785}{2\times1.9459}=\frac{6.785}{3.8918}\approx1.743 $$

Answer:

In terms of natural logarithms, \(x = \frac{\ln(889)}{2\ln(7)}\); the decimal approximation is \(x\approx1.743\)