Understanding the Logarithm Problem

The question asks us to find the value of $\log_b(ab)$ given that $\log_a(ab) = x$. This problem involves manipulating logarithmic expressions and changing the base of a logarithm.

We are given:

$\log_a(ab) = x$

We need to find the value of:

$\log_b(ab)$

Key Logarithm Properties

To solve this problem, we will use the following fundamental properties of logarithms:

• Product Rule: $\log_c(MN) = \log_c(M) + \log_c(N)$
• Logarithm of Base: $\log_c(c) = 1$
• Change of Base Formula: $\log_c(d) = \frac{\log_k(d)}{\log_k(c)}$ (for any suitable base k). A common case is $\log_c(d) = \frac{1}{\log_d(c)}$.

Step-by-Step Solution

Let's start with the given equation: $\log_a(ab) = x$.

Step 1: Apply the Product Rule to the given equation.

Using the product rule, $\log_a(ab)$ can be written as $\log_a(a) + \log_a(b)$.

So, the given equation becomes:

$\log_a(a) + \log_a(b) = x$

Step 2: Simplify using the Logarithm of Base property.

We know that $\log_a(a) = 1$. Substituting this into the equation:

$1 + \log_a(b) = x$

Step 3: Isolate $\log_a(b)$.

Subtracting 1 from both sides, we get:

$\log_a(b) = x - 1$

This gives us a relationship between the bases a and b.

Step 4: Now consider the expression we need to find: $\log_b(ab)$.

Apply the Product Rule to $\log_b(ab)$:

$\log_b(ab) = \log_b(a) + \log_b(b)$

Step 5: Simplify using the Logarithm of Base property.

We know that $\log_b(b) = 1$. Substituting this into the expression:

$\log_b(ab) = \log_b(a) + 1$

Step 6: Relate $\log_b(a)$ to $\log_a(b)$.

Using the change of base formula, we know that $\log_b(a) = \frac{1}{\log_a(b)}$.

Step 7: Substitute the value of $\log_a(b)$ from Step 3 into the expression for $\log_b(a)$.

We found $\log_a(b) = x - 1$. So,

$\log_b(a) = \frac{1}{x - 1}$

Step 8: Substitute the value of $\log_b(a)$ from Step 7 into the expression from Step 5.

We have $\log_b(ab) = \log_b(a) + 1$. Substituting $\log_b(a) = \frac{1}{x-1}$:

$\log_b(ab) = \frac{1}{x - 1} + 1$

Step 9: Simplify the expression.

To simplify, find a common denominator, which is $(x-1)$.

$\frac{1}{x - 1} + 1 = \frac{1}{x - 1} + \frac{x - 1}{x - 1}$

Combine the numerators over the common denominator:

$\frac{1 + (x - 1)}{x - 1} = \frac{1 + x - 1}{x - 1} = \frac{x}{x - 1}$

Thus, $\log_b(ab) = \frac{x}{x - 1}$.

Final Answer

Based on our calculations, if $\log_a(ab) = x$, then $\log_b(ab) = \frac{x}{x - 1}$.

Revision Table: Logarithm Properties

Additional Information: Understanding Logarithm Bases

The base of a logarithm tells us which number is being raised to a power. For example, $\log_{10}(100)$ asks "10 to what power equals 100?". The answer is 2, so $\log_{10}(100) = 2$. Similarly, $\log_2(8)$ asks "2 to what power equals 8?". The answer is 3, so $\log_2(8) = 3$.

The change of base formula is very useful because it allows us to convert a logarithm with an inconvenient base to one with a more convenient base (like base 10 or base e, which are available on calculators). The formula $\log_c(d) = \frac{\log_k(d)}{\log_k(c)}$ shows that the ratio of logs with a common base 'k' gives the log in base 'c'. A special case is when we swap the base and the argument, $\log_c(d) = \frac{1}{\log_d(c)}$, which is what we used to relate $\log_a(b)$ and $\log_b(a)$.