Problem 71 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } \log _{a}(a b)=x, \text { then evaluate } \log _{b}(a b) \text { in terms of } x \text { . } $$

Step-by-Step Solution

Answer

In terms of x, \(\log_{b}(ab) = 1\).

Step 1: Write down the given information

We are given that \(\log_{a}(ab) = x\).

Step 2: Use the change of base formula

With the change of base formula, we can rewrite the second logarithm in terms of the first base: \[\log_{b}(ab) = \frac{\log_{a}(ab)}{\log_{a}(b)}\]

Step 3: Substitute x

Since we know that \(\log_{a}(ab) = x\), we can substitute this into the formula from step 2: \[\log_{b}(ab) = \frac{x}{\log_{a}(b)}\]

Step 4: Rewrite the denominator in terms of x and a

To find the relationship between the denominator and x, we'll use the change of base formula again: \[\log_{a}(b) = \frac{\log_{a}(ab)}{\log_{a}(a)}\]

Step 5: Simplify the denominator

Since we know that \(\log_{a}(a) = 1\), the above equation simplifies to: \[\log_{a}(b) = \frac{\log_{a}(ab)}{1} = \log_{a}(ab)\]

Step 6: Substitute x again

Since we know that \(\log_{a}(ab) = x\), we can substitute this into the formula from step 5: \[\log_{a}(b) = x\]

Step 7: Substitute the denominator of the result in step 3

We found that \(\log_{a}(b) = x\), so we can substitute this back into the formula from step 3: \[\log_{b}(ab) = \frac{x}{x}\]

Step 8: Simplify the result

Finally, we see that the fraction simplifies to: \[\log_{b}(ab) = 1\] So, in terms of x, \(\log_{b}(ab) = 1\).

Key Concepts

Change of Base Formula

Logs can be tricky, but the change of base formula is a super handy tool! Imagine you have a log with a base that isn't too familiar. The change of base formula helps convert this log into a format that's more understandable.
To use it, you turn one logarithm into a fraction of two simpler logs. For example, \[\log_{b}(x) = \frac{\log_{a}(x)}{\log_{a}(b)}\]This formula is like translating a language—changing the base makes it easier to handle different problems.
In the given exercise, we needed to change the base from \(b\) to \(a\), which turned out very straightforward once we applied this marvelous formula.

Simplifying Logarithms

Logarithms might seem complex at first, but simplifying them can make things a lot easier. It’s like decluttering a messy room—it looks more organized and is easier to deal with!
In the exercise, after substituting the known values, the challenge was to simplify the expression.
We started with \(\log_{b}(ab) = \frac{x}{\log_{a}(b)}\), and by using known properties, we found that \(\log_{a}(b)\) equaled \(x\).
By substituting \(x\) into the equation, the complexity fell away, leaving us with a clean and simple result: \(1\).

• Simplification often requires substituting known values, which unveils the beauty of logs.
• Always keep an eye out for expressions that can cancel each other out!

Properties of Logarithms

Understanding the properties of logarithms unlocks a whole new level of problem-solving! These properties help you to manipulate and simplify logarithmic expressions efficiently.
For instance, one useful property is that \(\log_{a}(a) = 1\). Why? Because when a number is raised to 1, it equals itself.
This property simplified our expression considerably in the exercise. When we realized that \(\log_{a}(a)\) equals 1, it allowed us to see that \(\log_{a}(b)\) becomes simply \(x\).

• Properties like these reduce complex expressions into simpler ones.
• With some practice, these properties become intuitive and make solving log problems a breeze!
• Always remember, logs are just exponents in disguise!