Solving for x in Logarithmic Equation
We are given the equation: $ \log_x \left( \frac{5}{7} \right) = -\frac{1}{3} $ To find the value of x, we need to convert this logarithmic equation into its equivalent exponential form.
Logarithmic to Exponential Conversion
- The general rule is: If $ \log_b a = c $, then $ b^c = a $.
- Applying this rule to our equation, where $ b = x $, $ a = \frac{5}{7} $, and $ c = -\frac{1}{3} $, we get: $ x^{-1/3} = \frac{5}{7} $
Calculating the Value of x
- To solve for x, we raise both sides of the equation to the power of $ -3 $: $ \left( x^{-1/3} \right)^{-3} = \left( \frac{5}{7} \right)^{-3} $
- This simplifies to: $ x^1 = \left( \frac{7}{5} \right)^{3} $
- Now, we calculate the final value: $ x = \frac{7^3}{5^3} $ $ x = \frac{343}{125} $
Therefore, the value of $ x $ is $ \frac{343}{125} $.
Important Questions from Logarithms
- Real numbers $y$, $p$, and $n$ (all greater than 1) satisfy
$$(\log_{p^{1/n}} y)(\log_{y^{1/n}} p) = 16,$$
where the logarithms are taken to the bases $p^{1/n}$ and $y^{1/n}$.
The value of $n$ is ________ Consider two distinct positive real numbers $m, n$, with $m > n$.
Let $x = n^{\log_{10}(m)}$ and $y = m^{\log_{10}(n)}$. The relation between $x$ and $y$ is _______.
- A value of x that satisfies the equation $ \log x + \log (x - 7) = \log (x + 11) + \log 2 $ is
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 $x$ isLet $a = 30!$, $b = 50!$, and $c = 100!$. Consider the following numbers:
$log_a c$, $log_c a$, $log_b a$, $log_a b$
Which one of the following inequalities is CORRECT?
