Questions: Solve the logarithmic equation algebraically. Approximate the result to three decimal places. (If there is no solution, enter NO SOLUTION
5 log (x-5)=12
x=163.489 x
Transcript text: Solve the logarithmic equation algebraically. Approximate the result to three decimal places. (If there is no solution, enter NO SOLUTION
\[
\begin{array}{r}
5 \log (x-5)=12 \\
x=163.489 x
\end{array}
\]
Solution
Solution Steps
To solve the logarithmic equation \(5 \log (x-5) = 12\), we first isolate the logarithmic expression and then convert the logarithmic equation into its exponential form. This will allow us to solve for \(x\). Finally, we check if the solution is valid by ensuring the argument of the logarithm is positive.
Step 1: Isolate the Logarithmic Expression
Starting with the equation:
\[
5 \log (x - 5) = 12
\]
we divide both sides by 5 to isolate the logarithmic term:
\[
\log (x - 5) = \frac{12}{5}
\]
Step 2: Convert to Exponential Form
Next, we convert the logarithmic equation into its exponential form:
\[
x - 5 = 10^{\frac{12}{5}}
\]
Step 3: Solve for \(x\)
Now, we solve for \(x\) by adding 5 to both sides:
\[
x = 10^{\frac{12}{5}} + 5
\]
Calculating \(10^{\frac{12}{5}}\) gives approximately \(251.189\). Therefore:
\[
x \approx 251.189 + 5 = 256.189
\]
Step 4: Validate the Solution
We check if the solution is valid by ensuring that the argument of the logarithm is positive:
\[
x - 5 = 256.189 - 5 = 251.189 > 0
\]
Since the argument is positive, the solution is valid.