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Solve for [tex]\( x \)[/tex].

[tex]\[ 2^{5x - 1} = 3^x \][/tex]


Sagot :

To solve the equation [tex]\( 2^{5x - 1} = 3^x \)[/tex], follow these steps:

1. Understand the nature of the equation:
We have an equation involving exponential functions with different bases, [tex]\(2\)[/tex] and [tex]\(3\)[/tex].

2. Transform the equation using logarithms:
To handle the exponents with different bases, we apply logarithms (natural logarithms, for simplicity):
[tex]\[ \ln(2^{5x - 1}) = \ln(3^x) \][/tex]

3. Use logarithmic properties to simplify:
Apply the property of logarithms that allows you to bring down the exponent:
[tex]\[ (5x - 1) \ln(2) = x \ln(3) \][/tex]

4. Rearrange to isolate [tex]\(x\)[/tex]:
Let's distribute the logarithm terms and solve for [tex]\(x\)[/tex]:
[tex]\[ 5x \ln(2) - \ln(2) = x \ln(3) \][/tex]
[tex]\[ 5x \ln(2) - x \ln(3) = \ln(2) \][/tex]

5. Factor [tex]\(x\)[/tex] from the left-hand side:
[tex]\[ x (5 \ln(2) - \ln(3)) = \ln(2) \][/tex]

6. Solve for [tex]\(x\)[/tex]:
Divide both sides of the equation by [tex]\((5 \ln(2) - \ln(3))\)[/tex]:
[tex]\[ x = \frac{\ln(2)}{5 \ln(2) - \ln(3)} \][/tex]

7. Express in a simpler form:
Recognize that the solution can be expressed in terms of another logarithm. Here, we simplify:
[tex]\[ x = \log_{32/3}(2) \][/tex]

Note that [tex]\(\log_{32/3}(2)\)[/tex] can be interpreted using the change of base formula, but it effectively captures the solution in a compact form.

The precise solution is:
[tex]\[ x = \log_{32/3}(2) \][/tex]