Question

$log_2log_3e^{ln5^{log_57^{log_710^{log_{10}(8x-3)}}}}=13.is$

Answer 1

x = \frac{3^{8192}+3}{8}

Answer 2

Step-by-Step Solution

1. Simplify the Expression:

   The given equation is

   $$\log_2\left(\log_3\Big(e^{\ln\Big(5^{\log_5\Big(7^{\log_7\Big(10^{\log_{10}(8x-3)}\Big)}\Big)}\Big)}\Big)\right) = 13.$$

2. Remove the Exponential and Logarithm Pair:

   Notice that $e^{\ln A} = A$. Thus,

   $$e^{\ln\left(5^{\log_5\left(7^{\log_7\left(10^{\log_{10}(8x-3)}\right)}\right)}\right)} = 5^{\log_5\left(7^{\log_7\left(10^{\log_{10}(8x-3)}\right)}\right)}.$$

3. Apply the Inverse Property of Logarithms:

   Using the property $a^{\log_a(B)} = B$:

   $$5^{\log_5\left(7^{\log_7\left(10^{\log_{10}(8x-3)}\right)}\right)} = 7^{\log_7\left(10^{\log_{10}(8x-3)}\right)} = 10^{\log_{10}(8x-3)} = 8x-3.$$

4. Rewrite the Equation:

   The equation now reduces to

   $$\log_2\left(\log_3(8x-3)\right) = 13.$$

5. Solve for $\log_3(8x-3)$:

   Exponentiating both sides with base 2 gives:

   $$\log_3(8x-3) = 2^{13} = 8192.$$

6. Solve for $8x-3$:

   Exponentiating both sides with base 3:

   $$8x-3 = 3^{8192}.$$

7. Solve for $x$:

   $$8x = 3^{8192} + 3 \quad \Longrightarrow \quad x = \frac{3^{8192} + 3}{8}.$$

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Core Explanation:
Successively simplify the expression using the properties $e^{\ln A}=A$ and $a^{\log_a B}=B$. Reduce the original logarithmic equation to $\log_2(\log_3(8x-3))=13$, and then solve step-by-step using exponentiation.