QQuestionMathematics
QuestionMathematics
# Solve for $x$.
2^{x+ 5}= 13^{2 x}
Write the exact answer using either base- 10 or base-e logarithms.
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Answer
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Step 1:: To solve the equation, we will take the base- 10 logarithm (log) of both sides.
This is a valid operation since logarithms and exponentials are inverse operations. \log_{10}\left(2^{x+ 5}\right) = \log_{10}\left(13^{2x}\right)
Step 2:: Using the power rule for logarithms, which states that 1$, we can rewrite the equation as:
(x+ 5)\log_{10}(2) = 2x\log_{10}(13)
Step 3:: Next, we will isolate the term with 1$ on one side of the equation.
To do this, we can divide both sides by 1$.
Step 4:: Now, we can simplify the equation further by bringing all terms involving 1$ to the left side and constants to the right side.
Step 5:: Factor out 1$ from the left side of the equation.
Step 6:: Solve for 1$ by dividing both sides by the expression in parentheses.
Step 7:: Simplify the equation by calculating the value of the logarithm and performing the division.
x \approx - 5.83
Final Answer
The solution to the equation 1$.
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