Solving Logarithmic Equations: Finding the Value of m

The question asks us to find the value of 'm' in the given logarithmic equation:

$\log_8 m + \log_8 \frac{1}{6} = \frac{2}{3}$

To solve this equation, we can use the properties of logarithms. One key property is the product rule of logarithms, which states that $\log_b x + \log_b y = \log_b (xy)$.

Applying Logarithm Properties

Using the product rule on the left side of the equation, we can combine the two logarithmic terms:

$\log_8 \left( m \times \frac{1}{6} \right) = \frac{2}{3}$

$\log_8 \left( \frac{m}{6} \right) = \frac{2}{3}$

Converting to Exponential Form

Now we have a single logarithm. The definition of a logarithm states that if $\log_b a = c$, then this is equivalent to the exponential form $b^c = a$. In our equation, the base is 8, the exponent is $\frac{2}{3}$, and the argument is $\frac{m}{6}$.

Applying the definition, we convert the logarithmic equation into an exponential equation:

$8^{\frac{2}{3}} = \frac{m}{6}$

Evaluating the Exponential Term

To evaluate $8^{\frac{2}{3}}$, we can rewrite $8$ as $2^3$. Then we use the property of exponents $(a^x)^y = a^{xy}$.

$8^{\frac{2}{3}} = (2^3)^{\frac{2}{3}} = 2^{3 \times \frac{2}{3}} = 2^2$

So, $8^{\frac{2}{3}} = 4$.

Solving for m

Substitute the value back into the equation:

$4 = \frac{m}{6}$

To find 'm', multiply both sides of the equation by 6:

$m = 4 \times 6$

$m = 24$

Verifying the Solution

Let's check if $m=24$ satisfies the original equation:

$\log_8 24 + \log_8 \frac{1}{6}$

Using the product rule in reverse:

$\log_8 \left( 24 \times \frac{1}{6} \right) = \log_8 \left( \frac{24}{6} \right) = \log_8 4$

Now we need to check if $\log_8 4 = \frac{2}{3}$. Let $x = \log_8 4$. In exponential form, this is $8^x = 4$.

We can write both 8 and 4 with the same base, 2:

$(2^3)^x = 2^2$

$2^{3x} = 2^2$

Since the bases are equal, the exponents must be equal:

$3x = 2$

$x = \frac{2}{3}$

So, $\log_8 4 = \frac{2}{3}$, which matches the right side of the original equation. Thus, $m=24$ is the correct solution.

The value of m that satisfies the equation is 24.

Revision Table: Logarithm Properties Review

Additional Information: Understanding Fractional Exponents

A fractional exponent like $a^{p/q}$ can be understood in terms of roots and powers. The denominator 'q' indicates the root, and the numerator 'p' indicates the power. So, $a^{p/q} = \sqrt[q]{a^p} = (\sqrt[q]{a})^p$.

In our problem, $8^{2/3}$: The denominator is 3, meaning the cube root. The numerator is 2, meaning the square. So, $8^{2/3} = (\sqrt[3]{8})^2$.

• First, find the cube root of 8: $\sqrt[3]{8} = 2$, because $2^3 = 8$.
• Then, square the result: $2^2 = 4$.

This confirms that $8^{2/3} = 4$, as used in the solution process for the logarithmic equation.