For a real number $x > 1$,
$\frac{1}{\log_2 x} + \frac{1}{\log_3 x} + \frac{1}{\log_4 x} = 1$
The value of $x$ is
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
- 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 $
- 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 $
- Convert to exponential form: Recall that $ \log_b a = c $ is equivalent to $ b^c = a $. Applying this definition: $ x^1 = 24 $
- Determine the value of x: From the exponential form, we find: $ x = 24 $ The obtained value $x=24$ satisfies the initial condition $x > 1$.
Important Questions from Logarithms
For positive non-zero real variables $p$ and $q$, if
$\log (p^2 + q^2) = \log p + \log q + 2 \log 3$,
then, the value of $\frac{p^4+q^4}{p^2q^2}$ isA petrified wood fossil was discovered with 8 g of $^{14}C$. The decay of $^{14}C$ over time is given by:
$N_T = N_0 e^{-0.0001216T}$
If the half-life of $^{14}C$ is 5700 years, and the fossil initially had 32 g of $^{14}C$, the age of the fossil in years is ______.
- For integers $a$, $b$ and $c$, what would be the minimum and maximum values respectively of $a + b + c$ if $\log |a| + \log |b| + \log |c| = 0$?
- The value of the expression $\frac{1}{1+\log_u vw} + \frac{1}{1+\log_v wu} + \frac{1}{1+\log_w uv}$ is
- If $\log_x (5/7) = -1/3$, then the value of $x$ is
