Question

what is the solution to \log{25}x = 3?\
\bigcirc\\ x = \frac{3}{25}\
\bigcirc\\ x = \frac{8}{25}\
\bigcirc\\ x = 35\
\bigcirc\\ x = 40

Explanation:

Step1: Recall Logarithm Definition

The logarithmic equation \(\log_{a}b = c\) is equivalent to \(a^{c}=b\) (by the definition of logarithms, where \(a>0,a
eq1\)). Here, the equation is \(\log_{25}x = 3\), so \(a = 25\), \(c=3\), and \(b=x\).

Step2: Apply the Definition

Using the equivalence, we rewrite the logarithmic equation as an exponential equation: \(25^{3}=x\). Wait, but none of the options match \(25^{3}\). Wait, maybe the original equation is \(\log(25x)=3\) (if the base is 10, since it's not specified, usually base 10 for \(\log\) without a base). Let's assume base 10. Then \(\log_{10}(25x)=3\). By the definition of logarithm, \(10^{3}=25x\).

Step3: Solve for x

We know that \(10^{3}=1000\), so the equation becomes \(1000 = 25x\). Then, solve for \(x\) by dividing both sides by 25: \(x=\frac{1000}{25}=40\).

Answer:

x = 40 (the option with x = 40)