Question

question 24 of 24 step 1 of 1 solve the following exponential equation. express your answer as both an exact expression and a decimal approximation rounded to two decimal places. 8^(x+1) = 1684 answer 2 points x = u

Explanation:

Step1: Take log of both sides

Take the natural logarithm of both sides of the equation \(8^{x + 1}=1684\). We get \(\ln(8^{x + 1})=\ln(1684)\).
Using the logarithm power rule \(\ln(a^b)=b\ln(a)\), the left - hand side becomes \((x + 1)\ln(8)=\ln(1684)\).

Step2: Solve for \(x\)

First, divide both sides of the equation \((x + 1)\ln(8)=\ln(1684)\) by \(\ln(8)\):
\(x + 1=\frac{\ln(1684)}{\ln(8)}\)
Then, subtract 1 from both sides to solve for \(x\):
\(x=\frac{\ln(1684)}{\ln(8)}-1\)
Now, calculate the decimal approximation. We know that \(\ln(1684)\approx7.43\), \(\ln(8)\approx2.08\)
\(\frac{\ln(1684)}{\ln(8)}\approx\frac{7.43}{2.08}\approx3.57\)
\(x\approx3.57 - 1=2.57\)

Answer:

Exact form: \(x=\frac{\ln(1684)}{\ln(8)}-1\)
Decimal approximation: \(x\approx2.57\)