Solve for $x$.
$11^{-x-5}=18^{8 x}$
Write the exact answer using either base-10 or base-e logarithms.
Answer (2)
Take the natural logarithm of both sides: ln ( 1 1 − x − 5 ) = ln ( 1 8 8 x ) .
Apply the power rule: ( − x − 5 ) ln ( 11 ) = 8 x ln ( 18 ) .
Expand and rearrange: − x ln ( 11 ) − 5 ln ( 11 ) = 8 x ln ( 18 ) ⟹ − 5 ln ( 11 ) = x ( 8 ln ( 18 ) + ln ( 11 )) .
Solve for x : x = 8 l n ( 18 ) + l n ( 11 ) − 5 l n ( 11 ) .
\boxed{x = \frac{-5 \ln(11)}{8 \ln(18) + \ln(11)}}$ ### Explanation 1. Understanding the Problem We are given the equation $11^{-x-5}=18^{8 x}$. Our goal is to solve for $x$ and express the answer using either base-10 or base-e logarithms. 2. Applying Logarithms Take the natural logarithm (base-e) of both sides of the equation: \ln(11^{-x-5}) = \ln(18^{8x}) 3. Using the Power Rule Apply the power rule of logarithms, which states that $\ln(a^b) = b \ln(a)$: (-x-5)\ln(11) = 8x \ln(18) 4. E x p an d in g t h e Eq u a t i o n E x p an d t h e l e f t s i d eo f t h ee q u a t i o n : -x \ln(11) - 5 \ln(11) = 8x \ln(18) 5. Isolating x Terms Move all terms containing $x$ to one side of the equation: -5 \ln(11) = 8x \ln(18) + x \ln(11) 6. Factoring Factor out $x$ from the right side: -5 \ln(11) = x(8 \ln(18) + \ln(11)) 7. Solving for x Solve for $x$ by dividing both sides by $(8 \ln(18) + \ln(11))$: x = \frac{-5 \ln(11)}{8 \ln(18) + \ln(11)} 8. Approximating the Value of x We can approximate the value of $x$ using a calculator: x \approx \frac{-5 \times 2.3979}{8 \times 2.8904 + 2.3979} \approx \frac{-11.9895}{23.1232 + 2.3979} \approx \frac{-11.9895}{25.5211} \approx -0.4698 9. Final Answer The exact answer for $x$ is: x = \frac{-5 \ln(11)}{8 \ln(18) + \ln(11)}
Examples
Logarithms are incredibly useful in many real-world scenarios, such as calculating the magnitude of earthquakes using the Richter scale, determining the pH levels of chemical solutions, and modeling population growth. In finance, logarithms help in calculating continuously compounded interest rates. For instance, if you invest money at a continuously compounded interest rate, logarithms can help you determine how long it will take for your investment to double. This mathematical tool provides a way to understand and quantify phenomena that grow or decay exponentially.
To solve the equation 1 1 − x − 5 = 1 8 8 x , we take the natural logarithm of both sides and apply the power rule. After rearranging and isolating x , we find x = 8 l n ( 18 ) + l n ( 11 ) − 5 l n ( 11 ) .
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