Problem 32 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(N=6^{\log _{10} 40} .5^{\log _{10} 36}\), then find the value of \(N+10\).

Step-by-Step Solution

Answer

The value of \(N+10\) is 1450.

Step 1: Simplify the expression for N

Apply \(b^{log_bx}=x\) i.e the base \(b\) of the logarithm and the base \(b\) of the power are the same. So, it simplifies to \(N=40*36=1440\).

Step 2: Calculate the value of N+10

After simplifying the value of \(N\), add 10 to it i.e \(N+10=1440+10=1450\).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions wherein a constant base is raised to a variable exponent. These functions are central to multiple areas in mathematics due to their properties of rapid growth or decay.
Exponential expressions generally take the form of \(a^x\), where \(a\) is a constant known as the base, and \(x\) is the exponent or power.
Such functions are widely applied in growth and decay processes like population growth, radioactive decay, and compound interest.

• The function \(b^{\log_b x} = x\) is a useful identity that simplifies the expression when the base of the exponent matches the base of the logarithm.
• An exponential function can often be rearranged or simplified using logarithmic identities.

Logarithmic Identities

Logarithmic identities are rules that simplify expressions involving logarithms, making complex calculations more feasible. They are based on the properties of logarithms, which serve as the inverse operations of exponential functions.
Some key identities include:

• \(\log_b(xy) = \log_b x + \log_b y\)
• \(\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y\)
• \(\log_b(x^n) = n \log_b x\)
• \(b^{\log_b x} = x\), which was particularly used in the original problem to simplify the expression, as the base \(b\) and the log base match, allowing the exponent's logarithmic base to cancel out.

These identities were crucial in the simplification seen in the original solution, where matching bases led to the recognition that \(6^{\log_{10} 40}\) simplifies to 40, and \(5^{\log_{10} 36}\) simplifies to 36.

Simplification Techniques

Simplification techniques in the context of logarithms and exponential functions often involve using identities to reduce complex expressions into more manageable forms.
In the given problem, simplification was achieved using the identity \(b^{\log_b x} = x\), effectively reducing it to basic arithmetic.

• First, recognize where logarithmic identities can apply, as with \(\log_b x\) and the corresponding base \(x\).
• Another simplification trick involves breaking down expressions to see if identities apply, as seen with different bases in exponential forms.
• Simplifying expressions step-by-step helps avoid confusion and errors in larger problems, making it easier to evaluate final results accurately.

By using these strategies effectively, students can tackle intricate expressions with greater ease and accuracy, similar to how the solution's steps unraveled the problem to find the value of \(N\).