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
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 $\frac{p^4+q^4}{p^2q^2}$ isFor a real number $x > 1$,
$\frac{1}{\log_2 x} + \frac{1}{\log_3 x} + \frac{1}{\log_4 x} = 1$
The value of $x$ 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
