Math  /  Algebra

QuestionUse logarithms to solve. Give an exact simplified answer. Enter DNE if there is no solution. 5e8x=11x=\begin{array}{l} 5 e^{8 x}=11 \\ x=\square \end{array}
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Studdy Solution

STEP 1

What is this asking? Find the value of xx that makes 55 times ee to the power of 8x8x equal to 1111. Watch out! Remember the properties of logarithms and exponentials, and don't mix up the order of operations!

STEP 2

1. Isolate the exponential term
2. Apply the natural logarithm
3. Solve for xx

STEP 3

We want to get e8xe^{8x} by itself.
To do that, we need to **divide** both sides of the equation 5e8x=115e^{8x} = 11 by 5\textbf{5}.
This gives us: 5e8x5=115 \frac{5e^{8x}}{5} = \frac{11}{5} e8x=115 e^{8x} = \frac{11}{5} Now we have the exponential term all by itself!

STEP 4

Why the natural logarithm?
Because it's the inverse of the exponential function with base ee!
Applying the natural logarithm (ln) to both sides gives us: ln(e8x)=ln(115) \ln(e^{8x}) = \ln\left(\frac{11}{5}\right)

STEP 5

Remember the super cool property of logarithms: ln(ab)=bln(a)\ln(a^b) = b \cdot \ln(a).
Let's use it! 8xln(e)=ln(115) 8x \cdot \ln(e) = \ln\left(\frac{11}{5}\right) Since ln(e)\ln(e) is just 1\textbf{1} (because ln(e)\ln(e) means "e to what power gives me e?" and the answer is **1**), we have: 8x1=ln(115) 8x \cdot 1 = \ln\left(\frac{11}{5}\right) 8x=ln(115) 8x = \ln\left(\frac{11}{5}\right)

STEP 6

Almost there!
We just need to **divide** both sides by 8\textbf{8} to get xx by itself: 8x8=ln(115)8 \frac{8x}{8} = \frac{\ln\left(\frac{11}{5}\right)}{8} x=ln(115)8 x = \frac{\ln\left(\frac{11}{5}\right)}{8} And there we have it!

STEP 7

The exact simplified solution for xx is ln(115)8\frac{\ln\left(\frac{11}{5}\right)}{8}.

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