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Answer :
Certainly! To solve the equation [tex]\(4^{3x} = 37.9\)[/tex], we need to find the value of [tex]\(x\)[/tex].
Here's a step-by-step breakdown:
1. Understand the Equation:
We start with the equation [tex]\(4^{3x} = 37.9\)[/tex].
2. Apply Logarithms:
To eliminate the exponent, take the natural logarithm of both sides:
[tex]\[
\ln(4^{3x}) = \ln(37.9)
\][/tex]
3. Use Logarithmic Identity:
We can use the property of logarithms that states [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]. Apply this to the left side:
[tex]\[
3x \cdot \ln(4) = \ln(37.9)
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
To isolate [tex]\(x\)[/tex], divide both sides by [tex]\(3 \cdot \ln(4)\)[/tex]:
[tex]\[
x = \frac{\ln(37.9)}{3 \cdot \ln(4)}
\][/tex]
5. Calculate:
Performing this calculation gives a result for [tex]\(x\)[/tex] of approximately 0.874.
Therefore, the value of [tex]\(x\)[/tex] is approximately 0.874.
Here's a step-by-step breakdown:
1. Understand the Equation:
We start with the equation [tex]\(4^{3x} = 37.9\)[/tex].
2. Apply Logarithms:
To eliminate the exponent, take the natural logarithm of both sides:
[tex]\[
\ln(4^{3x}) = \ln(37.9)
\][/tex]
3. Use Logarithmic Identity:
We can use the property of logarithms that states [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]. Apply this to the left side:
[tex]\[
3x \cdot \ln(4) = \ln(37.9)
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
To isolate [tex]\(x\)[/tex], divide both sides by [tex]\(3 \cdot \ln(4)\)[/tex]:
[tex]\[
x = \frac{\ln(37.9)}{3 \cdot \ln(4)}
\][/tex]
5. Calculate:
Performing this calculation gives a result for [tex]\(x\)[/tex] of approximately 0.874.
Therefore, the value of [tex]\(x\)[/tex] is approximately 0.874.
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