Question

Solve each equation. $$ \log _{5} 625=x $$

Short answer

x = 4

Step-by-step solution

Identify the logarithmic form

The given equation is \(\text{log}_{5} 625 = x \). This is in logarithmic form where 5 is the base, 625 is the argument, and x is the exponent.

Convert to exponential form

Rewrite the logarithmic equation in its equivalent exponential form. The equation \(\text{log}_{5} 625 = x\) can be written as \(5^x = 625\).

Express 625 as a power of 5

Identify the power of 5 that equals 625. Notice that \(625 = 5^4\).

Set up an equation

Since both sides of the equation \(5^x = 625\) have the same base (5), we can set the exponents equal to each other. Thus, \(x = 4\).

Solve for x

The value of x that satisfies the equation \(\text{log}_{5} 625 = x\) is \(x = 4\).

Key concepts

logarithmic form

In mathematics, a logarithmic form is a way to represent an equation involving a logarithm. Here, the logarithm tells us what power we need to raise a specific base to get another number. For example, in the given problem \(\text{log}_{5} 625 = x \), the base is 5, the argument is 625, and x is the exponent. This can be read as '5 raised to what power equals 625?'. Logarithmic form helps to answer such questions and is written as \(\text{log}_{b} a = x \), where b is the base, a is the argument, and x is the exponent.

exponential form

When dealing with logarithms, it’s often useful to convert them into exponential form. The exponential form of a logarithmic equation shows the same relationship but in terms of exponentiation. For the logarithmic equation \(\text{log}_{5} 625 = x \), converting to exponential form gives us \(\text{5}^x = 625\). This conversion helps to solve the equation by making the relationship between the base, exponent, and argument more explicit. Essentially, it transforms the log equation into a form that can be more straightforward to manipulate and solve.

solving equations with logarithms

Solving equations with logarithms often involves several steps. First, you need to identify the equation's logarithmic form. Next, convert the equation to its exponential form to make it easier to solve. In our example, \(\text{log}_{5} 625 = x\) is converted to \(\text{5}^x = 625\). To solve for x, express 625 as a power of 5. Notice that \(\text{625} = 5^4\). Since the bases are the same, you can set the exponents equal to each other. Thus, \(\text{x = 4}\). These steps break down the process and make it simpler to find the solution.

changing base

Changing the base of a logarithm is a useful technique, especially when dealing with complex equations. The change of base formula lets you rewrite a logarithm in terms of logs with a base that is easier to work with, typically 10 or e (natural logarithms). The formula for changing base is \(\text{log}_{b} a = \frac{\text{log}_{c} a}{\text{log}_{c} b}\), where c is the new base. This can simplify calculations and make solving logarithmic equations more manageable. Always remember, changing the base does not alter the value of the logarithm, it merely represents it in a different form.