Logarithm Equation Solving

The problem requires finding the value of $x$ for the equation:

$ \frac{1}{\log_2 x} + \frac{1}{\log_3 x} + \frac{1}{\log_4 x} = 1 $

We are given the condition $x > 1$. The solution involves applying key logarithm properties.

Logarithm Properties Applied

• Change of Base Formula: $ \frac{1}{\log_a b} = \log_b a $. This allows us to change the base of the logarithm to $x$.
• Sum Rule for Logarithms: $ \log_b m + \log_b n = \log_b (mn) $. This helps combine multiple logarithm terms with the same base.

Solution Steps

1. Transform the equation: Using the change of base formula, rewrite each term with base $x$: $ \log_x 2 + \log_x 3 + \log_x 4 = 1 $
2. Combine logarithmic terms: Apply the sum rule for logarithms: $ \log_x (2 \times 3 \times 4) = 1 $ Simplify the product inside the logarithm: $ \log_x 24 = 1 $
3. Convert to exponential form: Recall that $ \log_b a = c $ is equivalent to $ b^c = a $. Applying this definition: $ x^1 = 24 $
4. Determine the value of x: From the exponential form, we find: $ x = 24 $ The obtained value $x=24$ satisfies the initial condition $x > 1$.