Question

Solve each equation. $$ \log _{5} x=3 $$

Short answer

x = 125

Step-by-step solution

- Understand the Equation

First, recognize that the equation is a logarithmic equation: \ \ \ \(\log_{5} x = 3\)

- Rewrite in Exponential Form

Convert the logarithmic equation to its exponential form. The formula for conversion is \(\log_{a} b = c\) which converts to \(a^{c} = b\). Here, \(a = 5\), \(b = x\), and \(c = 3\). So, rewrite the equation as: \ \ \ \[5^3 = x\]

- Solve for x

Calculate the value of \(5^3\): \ \ \ \[5^3 = 5 \times 5 \times 5 = 125\] \ \ Therefore, \(x = 125\)

Key concepts

Exponential Form

Logarithmic equations can often seem confusing, but one of the simplest ways to understand and solve them is by converting them into exponential form. To do this, remember that a logarithm is essentially an exponent in disguise. If we have a logarithmic equation like \(\text{log}_a b = c\), we can convert it to its exponential form as \[a^c = b \]. This equivalence is useful because exponential equations are usually more straightforward to solve. For example, if we take \(\text{log}_{5} x = 3\) and convert it to exponential form, it becomes \[5^3 = x\]. This tells us that raising 5 to the power of 3 will give us the value of x.

Logarithms

A logarithm is a mathematical operation that helps us determine how many times a number (called the base) must be multiplied by itself to get another number. In the equation \(\text{log}_{5} x = 3\), 5 is the base, and we are looking for the value of x such that 5 raised to some power equals x. Specifically, we need to find the number of times we need to multiply 5 to get a certain value. \(\text{log}_{a} b = c\) answers the question, 'To what power do we need to raise a to obtain b?' Here's a quick reference: \(\text{log}_{10} 100 = 2\) because 10 raised to the power of 2 is 100. Similarly, \(\text{log}_{2} 8 = 3\), because 2 to the power of 3 is 8.

Solving Equations

The final step of solving logarithmic equations is to calculate the unknown value. After converting the logarithmic equation \(\text{log}_{5} x = 3\) to its exponential form \[5^3 = x\], we can proceed to evaluate the expression. This means calculating \[5^3\] which equals \[5 \times 5 \times 5 = 125.\] Therefore, the solution to the equation is \[x = 125.\] Solving logarithmic equations often becomes easier once they are rewritten in their exponential form. Practicing these conversions and calculations can significantly improve your problem-solving skills in logarithmic and exponential contexts.