Question
Find the solution of the exponential equation \( 14^{-\frac{z}{7}}=18 \) in terms of logarithms, or correct to four decimal places. \( x=\square \)
Ask by Hammond Henry. in the United States
Mar 13,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( z \approx -7.6666 \)
Solution
Solve the equation \( 14^{-\frac{z}{7}}=18 \).
Solve the equation by following steps:
- step0: Solve for \(z\):
\(14^{-\frac{z}{7}}=18\)
- step1: Take the logarithm of both sides:
\(\log_{14}{\left(14^{-\frac{z}{7}}\right)}=\log_{14}{\left(18\right)}\)
- step2: Evaluate the logarithm:
\(-\frac{z}{7}=\log_{14}{\left(18\right)}\)
- step3: Rewrite the expression:
\(\frac{-z}{7}=\log_{14}{\left(18\right)}\)
- step4: Cross multiply:
\(-z=7\log_{14}{\left(18\right)}\)
- step5: Change the signs:
\(z=-7\log_{14}{\left(18\right)}\)
The solution to the exponential equation \(14^{-\frac{z}{7}}=18\) in terms of logarithms is \(z=-7\log_{14}(18)\).
To correct to four decimal places, we can calculate the value of \(z\) as follows:
\(z \approx -7.666602\)
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Simplify this solution Mind Expander
To solve the equation \( 14^{-\frac{z}{7}}=18 \), we can use logarithms. First, take the natural logarithm of both sides: \[ -\frac{z}{7} \ln(14) = \ln(18) \] Now, isolate \( z \): \[ -\frac{z}{7} = \frac{\ln(18)}{\ln(14)} \] Multiplying both sides by -7 gives: \[ z = -7 \cdot \frac{\ln(18)}{\ln(14)} \] Calculating this gives: \[ z \approx -7 \cdot \frac{2.8904}{2.6391} \approx -7 \cdot 1.095 = -7.665 \] Thus, correct to four decimal places, \( z \approx -7.6650 \).
