Q. What is the true solution to \(2 \ln(4x) = 2 \ln(8)\)?
Answer
\[
2\ln(4x)=2\ln 8
\]
Divide both sides by 2:
\[
\ln(4x)=\ln 8
\]
Exponentiate:
\[
4x=8
\]
Thus
\[
x=2
\]
(and \(x>0\), so valid).
Detailed Explanation
Problem
Solve the equation \(2\ln(4x)=2\ln 8\).
Step-by-step solution
-
Divide both sides of the equation by 2 to simplify. This yields
\[\ln(4x)=\ln 8.\]
-
Use the one-to-one property of the natural logarithm: if \(\ln A=\ln B\) then \(A=B\), provided \(A>0\) and \(B>0\). Applying this gives
\[4x=8.\]
-
Solve the linear equation for \(x\):
\[x=\frac{8}{4}=2.\]
-
Check the domain condition for the logarithm. The argument of \(\ln(4x)\) must be positive, so \(4x>0\) which means \(x>0\). The found solution \(x=2\) satisfies this condition.
Answer
The solution is \(x=2\).
See full solution
FAQs
What is the true solution to 2 ln 4x = 2 ln 8?
Divide by 2: \(\ln(4x) = \ln(8)\). Then \(4x = 8\), so \(x = 2\). Check domain: \(4x>0\), so \(x>0\); \(x = 2\) is valid.
Why can I divide both sides by 2?
Because 2 is a nonzero constant. Dividing an equation by a nonzero number preserves equality: 2 ln(4x) / 2 = 2 ln 8 / 2 gives ln(4x) = ln 8.
Why does ln(4x) = ln 8 imply 4x = 8?
The natural log function ln is one-to-one on its domain (positive reals). If ln = ln B, then = B, so 4x = 8.
Could there be a negative solution like x = -2?
If you rewrite \(2 \ln(4x) = \ln((4x)^2)\) and \(2 \ln(8) = \ln(8^2)\), you might get \((4x)^2 = 64 \Rightarrow x = \pm 2\). But the log requires \(4x > 0\), so \(x > 0\); \(x = -2\) is extraneous. Only \(x = 2\) remains.
How else can I solve it using exponentiation?
Exponentiate both sides: e^{ln(4x)} = e^{ln 8} gives 4x = 8, so x = 2. Exponentiation with base e reverses the ln.
What domain restrictions should I check?
What domain restrictions should I check?
What if the equation had different log bases, like log_b?
The one-to-one property holds for any logarithm base \(b>0, b \neq 1\): \(\log_b(A) = \log_b(B)\) implies \(A = B\). The same solving steps apply.
Can I move the 2 inside the log as an exponent?
Yes: 2 ln(4x) = ln((4x)^2). That is valid, but be careful: squaring can introduce extraneous roots; always check the original domain after solving.
Solve 2 ln(4x) = 2 ln 8 for x today.
Check solution steps below.
Check solution steps below.
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