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Answer :
To solve the equation [tex]\(4^x = 128\)[/tex] without knowing the power of 4 that equals 128, you can use logarithms, which is a systematic approach. Here's how you can do it:
1. Take the logarithm of both sides: You can use any logarithm, but the common logarithm (base 10) or the natural logarithm (base e) are typically used. Here, we'll use the natural logarithm ("ln").
[tex]\[
\ln(4^x) = \ln(128)
\][/tex]
2. Apply the logarithm power rule: The power rule for logarithms states that [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]. So, apply this to the left side:
[tex]\[
x \cdot \ln(4) = \ln(128)
\][/tex]
3. Solve for x: Divide both sides by [tex]\(\ln(4)\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{\ln(128)}{\ln(4)}
\][/tex]
4. Calculate the values: Using a calculator, find the values of [tex]\(\ln(128)\)[/tex] and [tex]\(\ln(4)\)[/tex]:
[tex]\[
\ln(128) \approx 4.852
\][/tex]
[tex]\[
\ln(4) \approx 1.386
\][/tex]
5. Complete the division:
[tex]\[
x \approx \frac{4.852}{1.386} \approx 3.5
\][/tex]
So, the solution to the equation [tex]\(4^x = 128\)[/tex] is approximately [tex]\(x = 3.5\)[/tex].
Graphs and intersections or guessing calculations using multiplication won't necessarily give you a precise answer, especially if you don’t know an exact power of 4 that results in 128. Using logarithms effectively finds the exact exponent needed.
1. Take the logarithm of both sides: You can use any logarithm, but the common logarithm (base 10) or the natural logarithm (base e) are typically used. Here, we'll use the natural logarithm ("ln").
[tex]\[
\ln(4^x) = \ln(128)
\][/tex]
2. Apply the logarithm power rule: The power rule for logarithms states that [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]. So, apply this to the left side:
[tex]\[
x \cdot \ln(4) = \ln(128)
\][/tex]
3. Solve for x: Divide both sides by [tex]\(\ln(4)\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{\ln(128)}{\ln(4)}
\][/tex]
4. Calculate the values: Using a calculator, find the values of [tex]\(\ln(128)\)[/tex] and [tex]\(\ln(4)\)[/tex]:
[tex]\[
\ln(128) \approx 4.852
\][/tex]
[tex]\[
\ln(4) \approx 1.386
\][/tex]
5. Complete the division:
[tex]\[
x \approx \frac{4.852}{1.386} \approx 3.5
\][/tex]
So, the solution to the equation [tex]\(4^x = 128\)[/tex] is approximately [tex]\(x = 3.5\)[/tex].
Graphs and intersections or guessing calculations using multiplication won't necessarily give you a precise answer, especially if you don’t know an exact power of 4 that results in 128. Using logarithms effectively finds the exact exponent needed.
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