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What is the true solution to the equation below?

[tex]\[ 2 \ln(e^{\ln(2x)}) - \ln(e^{\ln(10x)}) = \ln(30) \][/tex]

A. [tex]\(x = 30\)[/tex]
B. [tex]\(x = 75\)[/tex]
C. [tex]\(x = 150\)[/tex]
D. [tex]\(x = 300\)[/tex]


Sagot :

Let's solve the given equation step-by-step, ensuring we cover each detailed aspect:

Given equation:
[tex]\[ 2 \ln\left(e^{\ln(2) \cdot x}\right) - \ln\left(e^{\ln(10) \cdot x}\right) = \ln(30) \][/tex]

Step 1: Simplify the logarithmic expressions using basic logarithmic properties.

We know that [tex]\(\ln(e^y) = y\)[/tex]. Applying this property:

[tex]\[ \ln\left(e^{\ln(2) \cdot x}\right) = \ln(2) \cdot x \][/tex]
[tex]\[ \ln\left(e^{\ln(10) \cdot x}\right) = \ln(10) \cdot x \][/tex]

Substituting these into the equation:

[tex]\[ 2 \cdot (\ln(2) \cdot x) - (\ln(10) \cdot x) = \ln(30) \][/tex]
[tex]\[ 2 \ln(2) \cdot x - \ln(10) \cdot x = \ln(30) \][/tex]

Step 2: Combine like terms.

Factor [tex]\(x\)[/tex] out from the terms on the left side:

[tex]\[ x(2 \ln(2) - \ln(10)) = \ln(30) \][/tex]

Step 3: Isolate [tex]\(x\)[/tex].

To solve for [tex]\(x\)[/tex], divide both sides by [tex]\(2 \ln(2) - \ln(10)\)[/tex]:

[tex]\[ x = \frac{\ln(30)}{2 \ln(2) - \ln(10)} \][/tex]

Step 4: Simplify the denominator using properties of logarithms.

We recognize that:

[tex]\[ \ln\left(2^2\right) = 2 \ln(2) \][/tex]

Then:

[tex]\[ 2 \ln(2) - \ln(10) = \ln(4) - \ln(10) \][/tex]

Using the property of logarithms that [tex]\(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\)[/tex]:

[tex]\[ \ln(4) - \ln(10) = \ln\left(\frac{4}{10}\right) = \ln\left(\frac{2}{5}\right) \][/tex]

Therefore, the expression for [tex]\(x\)[/tex] becomes:

[tex]\[ x = \frac{\ln(30)}{\ln\left(\frac{2}{5}\right)} \][/tex]

The expression [tex]\(\frac{\ln(30)}{\ln\left(\frac{2}{5}\right)}\)[/tex] can be further interpreted as [tex]\(\log_{\frac{2}{5}}(30)\)[/tex], but this interpretation steps into advanced logarithmic terms beyond the basics of solving the equation.

The result for [tex]\(x\)[/tex] is therefore:

[tex]\[ x = \log_\frac{2}{5}(30^{\frac{1}{\log(2/5)}}) \][/tex]

Given this, the correct solution to the original equation is not among the provided options (30, 75, 150, 300) directly without a specific transformation, but rather the result we obtained mathematically represents the exact solution:

[tex]\[ x = \log\left(30^{\frac{1}{\log\left(\frac{2}{5}\right)}}\right) \][/tex]

So, the exact solution to the equation is:

[tex]\[ x = \log\left(30^{\frac{1}{\log(2/5)}}\right) \][/tex]
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