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Solve for x in the equation 3^(x+2) = 2^(x+3).
Published on November 5, 2024
Algebra
The solution to the equation 3^(x+2) = 2^(x+3) is found by taking the natural logarithm of both sides, applying logarithm properties, isolating x, and simplifying to an expression involving natural logarithms of 2 and 3. The approximate value of x is then calculated.
Question
Answer
Here’s how to solve the equation
-
Take the logarithm of both sides. We can use any base, but the common logarithm (base 10) or the natural logarithm (base
) are usually easiest. Let’s use the natural logarithm: -
Use the power rule of logarithms. This rule states that
. Applying this rule to both sides of our equation gives: -
Expand both sides:
-
Isolate the terms with ‘x’ on one side and the constant terms on the other:
-
Factor out ‘x’ on the left side:
-
Divide both sides by
to solve for x: -
Simplify using logarithm properties:
-
Calculate the approximate value:
Therefore,
Final Answer: The final answer is