Problem 1 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve for \(x: x+\log _{10}\left(1+2^{x}\right)=x \log _{10} 5+\log _{10} 6\)

Step-by-Step Solution

Answer

The final solution after isolating for \(x\) is: \(x = \frac{10^{\log_{10} {6}} - \frac{1}{5}}{ \frac{2^x}{5}}\)

Step 1: Group Terms with x on One Side

First, let's rearrange the equation and bring all terms that include the variable \(x\) towards one side. The equation then becomes: \(x+\log_{10}(1+2^x) - x\log_{10} 5 = \log_{10} 6\)

Step 2: Split Logarithmic Terms

Next, since \(\log{a} - \log{b} = \log{\frac{a}{b}}\), we can simplify \(\log_{10}{(1+2^x)} - \log_{10}{5}\) : the equation transforms into \(x + \log_{10}{\left(\frac{1+2^x}{5}\right)} = \log_{10} {6}\)

Step 3: Apply Exponential Property

Now, applying the exponential property for the term inside the logarithm function, it can be rewritten as \(x + \log_{10}{\left(\frac{1}{5} + \frac{2^x}{5}\right)} = \log_{10} {6}\)

Step 4: Remove Logarithms

In the next step, remove the logarithms by raising both sides of the equation to the base 10. This gives us \(10^x \left(\frac{1}{5} + \frac{2^x}{5}\right) = 10^{\log_{10} {6}}\)

Step 5: Simplify

Finally, we can simplify the expression using the formula \(10^{\log_{10} {a}} = a\) and isolate for \(x\). This gives the final solution as \(x = \frac{10^{\log_{10} {6}} - \frac{1}{5}}{ \frac{2^x}{5}}\)

Key Concepts

Logarithmic Equations

Logarithmic equations are equations that involve logarithms of unknown values. An essential part of solving these equations is understanding the properties of logarithms.
When working with logarithmic equations, you should know certain operations make solving them easier:

• The property that allows subtraction of logs to become division inside a log: \(\log(a) - \log(b) = \log\left(\frac{a}{b}\right)\).
• Using the definition of logarithms: \(\log_b(a) = c\) effectively means \(b^c = a\).
• Proficiency in manipulating equations to group like terms together.

For a problem like the one given, the challenge is to separate the terms involving the variable, often by leveraging these logarithmic properties to simplify the equation.

Exponential Properties

Exponential properties are vital when solving equations involving logs and powers. The exponential property indicates how a base number raised to a power behaves.
The relevant properties often include:

• Each side of an equation involving logs can be raised as a power: If \(\log_b(x) = y\), then \(x = b^y\).
• Recognizing that \(10^{\log_{10}(a)} = a\); this helps simplify expressions where logs need removal.

In exercises such as these, after maneuvering the equation to isolate terms, apply the exponential property to eliminate logs and simplify the equation further. This step is crucial to transform from logs back into straightforward expressions that are more easily solved.

Solving Logarithms

Solving logarithms involves reversing the logarithmic format to find values for unknown terms. This requires an understanding of basic logarithm rules as well as the relationship between logs and exponentials.
Here's how you can solve equations with logs:

• Isolate the logarithmic part of the equation first. This often involves simplifying and rearranging terms.
• Use properties such as converting the addition and subtraction of logs into products and quotients: \(\log_b(m) + \log_b(n) = \log_b(m \cdot n)\).
• Once isolated, apply the inverse operation, typically raising the base of the log to remove the logarithmic form.

For the exercise provided, once the logs are isolated, use these reverse operations to find a useful mathematical form. This will help in solving for the variable efficiently by changing it to an exponential equation.