CAT Base Change rule - Practice Questions & MCQ

CAT 2024: Exam Date, Registration, Syllabus, Pattern, Preparation Tips, What is CAT Exam

Quick Facts

4 Questions around this concept.

Solve by difficulty

EASY

If x = log₂ 8 and y = log₄ 8, then $\frac{1}{x}+\frac{1}{y}=$

$\frac{1}{4}$

$\frac{1}{2}$

View Solution

Concepts Covered - 1

Base Change rule

The Base Change Rule is a powerful tool in logarithms, allowing you to change the base of a logarithmic expression. This rule is particularly useful when trying to simplify expressions or when working with a calculator that might only compute logarithms of a specific base.

Base Change Rule:

If you have a logarithm with base b and you want to change the base to a new base c , the formula is:

$\log _b a=\frac{\log _c a}{\log _c b}$

Where:

- $\log _b a$ is the logarithm of a with base b.

- $\log _c a$ is the logarithm of a with the new base c.

- $\log _c b$ is the logarithm of the original base b with the new base c.

Solved Examples:

1.Convert to $log_749$ base 2.

Solution:

Using the base change rule:

$\begin{aligned} \log _7 49 & =\frac{\log _2 49}{\log _2 7} \end{aligned}$

We know $7^{2}=49$ and thus $log_7 49=2$

Also, $2^{2}.807\approx 7$

Thus, $log7\approx 2.807$

Therefore, $log749\approx \frac{2}{2.807}\approx 0.712$ in base 2.

If , $log5 125=3$ find $log_5 25$ in terms of base 10.

Solution:

Using the base change rule:

$\mathrm{\begin{gathered} \log _5 25=\frac{\log _{10} 25}{\log _{10} 5} \\ A s \: 5^2=25, \log _{10} 25=2 \log _{10} 5 \\ \text { Thus, } \log _5 25=\frac{2}{\log _{10} 5} \end{gathered}}$

Tips and Tricks:

1. Direct Application: The Base Change Rule can often be directly applied to simplify logarithmic expressions, especially when trying to match bases with given information.

2. Calculator Usage: Most scientific calculators have keys for (common logarithm) and (natural logarithm). The Base Change Rule can be invaluable when you need to compute logarithms of other bases.

3. Simplify First: Before applying the Base Change Rule, check if the original logarithm can be simplified. This might reduce the number of steps needed.

4. Familiarise with Common Logarithm Values: Knowing logarithm values of certain numbers in specific bases (like 2, e, 10) can speed up the process.

5. Practice with Different Bases: Ensure you practise using the Base Change Rule with a variety of bases to be comfortable in any exam scenario.

In conclusion, the Base Change Rule is an essential concept in logarithms, offering flexibility in changing and simplifying bases. A thorough understanding of this rule, paired with regular practice, will equip students to tackle a wide array of logarithmic problems in management entrance exams.