Question

What is the value of x in the following expression?
\(x+\log_{10}(1+2^x)=x\log_{10}5+\log_{10}6\)

• A. 1
• B. 0
• C. -1
• D. 3

Answer 1

The Correct Option is A

To solve the equation \(x+\log_{10}(1+2^x)=x\log_{10}5+\log_{10}6\), we'll start by simplifying and analyzing the given expression. The equation we need to solve is:

\( x + \log_{10}(1 + 2^x) = x \log_{10} 5 + \log_{10} 6 \)

We can rearrange the expression as follows:

\( \log_{10}(1 + 2^x) - \log_{10} 6 = x \log_{10} 5 - x \)

This simplifies to:

\( \log_{10}\left(\frac{1 + 2^x}{6}\right) = x(\log_{10} 5 - 1) \)

Let us assume a value for \(x\) to verify which one satisfies the equation. Begin with the given options:

Option 1: \(x = 1\)

Substitute \(x = 1\) into the equation:

\( \log_{10}\left(\frac{1 + 2^1}{6}\right) = 1(\log_{10} 5 - 1) \)

Simplify:

\( \log_{10}\left(\frac{3}{6}\right) = \log_{10} 5 - 1 \)

\( \log_{10}\left(\frac{1}{2}\right) = \log_{10} 5 - \log_{10} 10 \)

Simplifying further:

\( -\log_{10} 2 = \log_{10} 5 - \log_{10} 10 \)

\( -0.3010 = 0.69897 - 1 \)

\( -0.3010 = -0.3010 \) \)

The equation holds true; therefore, x = 1 is indeed a solution.

Option 2: Trying other options (0, -1, 3) similarly will not satisfy the original equation upon substitution, thus confirming that the correct solution is \(x = 1\).