Q. what is the solution to 4\log_{4}(x+8)=4^{2}

Q: What does 4\log_4(x+8) mean?

It means multiplication: 4\log_4(x+8)=4\cdot\log_4(x+8). It is not \log_4(4(x+8)). Parentheses and placement matter.

Q: How do you solve 4\log_4(x+8)=4^2 step-by-step?

Divide by 4: \log_4(x+8)=4. Exponentiate base 4: x+8=4^4=256. So x=248. Check domain x>-8.

Q: Are there domain restrictions or extraneous solutions?

Yes: the argument of a log must be positive, so x+8>0\Rightarrow x>-8. The found solution x=248 satisfies this, so it is valid.

Q: How can you rewrite the equation using natural logs?

Use change of base: \log_4(x+8)=\dfrac{\ln(x+8)}{\ln 4}. Then 4\cdot\dfrac{\ln(x+8)}{\ln4}=16, solve for x to get x=248.

Q: How do I check the solution x=248?

Substitute: 4\log_4(248+8)=4\log_4(256)=4\cdot4=16. Right side 4^2=16. Both sides match.

Q: Is \log_4((x+8)^4) the same as 4\log_4(x+8)?

Yes, by the power rule: \log_4((x+8)^4)=4\log_4(x+8), provided x+8>0. They are equivalent expressions.

Q: What if the base weren't 4, say a\log_a(x+b)=a^2?

Divide by a to get \log_a(x+b)=a. Then x+b=a^a, so x=a^a-b. Require a>0, a\neq1, and x+b>0.

Q: Could the left side ever mean \log_4(4(x+8))?

Not typically; standard notation 4\log_4(x+8) denotes multiplication. \log_4(4(x+8)) would be written explicitly with parentheses. They are different functions.

Answer

Divide both sides by 4:
\[
\log_{4}(x+8)=4
\]
Exponentiate base 4:
\[
x+8=4^{4}=256
\]
So
\[
x=256-8=248
\]
Domain check: \(x>-8\), so \(x=248\) is valid.

Detailed Explanation

Write the given equation and recognize the structure.

\[4\log_{4}\bigl(x+8\bigr)=4^{2}\]

Interpretation: the left side is 4 times the logarithm base 4 of (x+8); the right side is 4 squared.
Isolate the logarithm by dividing both sides by 4.

\[\log_{4}\bigl(x+8\bigr)=\frac{4^{2}}{4}\]

Simplify the right side: \(\frac{4^{2}}{4}=\frac{16}{4}=4\). So

\[\log_{4}\bigl(x+8\bigr)=4\]
Convert the logarithmic equation to its equivalent exponential form.

The definition of \(\log_{4}\bigl(x+8\bigr)=4\) means

\[x+8=4^{4}\]

Compute \(4^{4}=256\), so

\[x+8=256\]
Solve for x by subtracting 8 from both sides.

\[x=256-8\]

\[x=248\]
Check the domain and verify the solution.

Logarithm requires \(x+8>0\), so \(x>-8\). The found value \(x=248\) satisfies this.

Optionally substitute back into the original equation to confirm:

Left side: \(4\log_{4}(248+8)=4\log_{4}(256)=4\cdot 4=16\). Right side: \(4^{2}=16\). They match.

Final answer: \(x=248\).

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FAQs

What does \(4\log_4(x+8)\) mean?

It means multiplication: \(4\log_4(x+8)=4\cdot\log_4(x+8)\). It is not \(\log_4(4(x+8))\). Parentheses and placement matter.

How do you solve \(4\log_4(x+8)=4^2\) step-by-step?

Divide by 4: \(\log_4(x+8)=4\). Exponentiate base 4: \(x+8=4^4=256\). So \(x=248\). Check domain \(x>-8\).

Are there domain restrictions or extraneous solutions?

Yes: the argument of a log must be positive, so \(x+8>0\Rightarrow x>-8\). The found solution \(x=248\) satisfies this, so it is valid.

How can you rewrite the equation using natural logs?

Use change of base: \(\log_4(x+8)=\dfrac{\ln(x+8)}{\ln 4}\). Then \(4\cdot\dfrac{\ln(x+8)}{\ln4}=16\), solve for \(x\) to get \(x=248\).

How do I check the solution \(x=248\)?

Substitute: \(4\log_4(248+8)=4\log_4(256)=4\cdot4=16\). Right side \(4^2=16\). Both sides match.

Is \(\log_4((x+8)^4)\) the same as \(4\log_4(x+8)\)?

Yes, by the power rule: \(\log_4((x+8)^4)=4\log_4(x+8)\), provided \(x+8>0\). They are equivalent expressions.

What if the base weren't 4, say \(a\log_a(x+b)=a^2\)?

Divide by \(a\) to get \(\log_a(x+b)=a\). Then \(x+b=a^a\), so \(x=a^a-b\). Require \(a>0\), \(a\neq1\), and \(x+b>0\).

Could the left side ever mean \(\log_4(4(x+8))\)?

Not typically; standard notation \(4\log_4(x+8)\) denotes multiplication. \(\log_4(4(x+8))\) would be written explicitly with parentheses. They are different functions.

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