Problem 62 - Precalculus : Building Concepts and Connections

Question

In Exercises \(53-64,\) simplify each expression. Assume that each variable expression is defined for appropriate values of \(x .\) Do not use a calculator. $$\log _{5} 625$$

Step-by-Step Solution

Answer

The simplified form of \(log_5(625)\) is \(4\).

Step 1: Identify the base and the number

Here, the base of the logarithm is \(5\) and the number we're taking the log of is \(625\). Our task is to figure out what power we would have to raise \(5\) to in order for it to equal \(625\). This can also be written as \(5^? = 625\).

Step 2: Find the exponent

We need to think of what power we can raise \(5\) to in order to get \(625\). From knowledge of basic powers, we know that \(5^4 = 625\). Thus, the '?' in the equation \(5^? = 625\) can be substituted with \(4\).

Step 3: Write the solution

Therefore, \(log_5 (625) = 4\), since \(5^4\) equals to \(625\) as determined in step 2. This means you would need to raise \(5\) to the power of \(4\) to get \(625\).

Key Concepts

Logarithmic Expressions

Understanding logarithmic expressions is crucial for solving a variety of mathematical problems. Essentially, a logarithm tells us how many times we need to multiply the base to reach a certain number. It's often stated as \( \log_b(a) \), where \(\b\) is the base and \(\b\) is the number we're looking into. In our exercise \( \log_5(625) \), we're about to find out how many fives must be multiplied together to produce 625.

Whenever you're faced with simplifying a logarithmic expression, start by identifying the base and the number. As visualized in the exercise, think of the expression in the form of an equation \(\b^x = \b\), where \(\b\) is the base and \(\b\) is the number. Next, search for a common exponent that would make the base equal to the number. The result of this is the log value you seek. Remembering multiplication facts or knowing the powers of certain numbers can greatly simplify this process.

The ultimate goal with logarithmic expressions is to express complex multiplications in a more manageable form, which can be incredibly helpful when dealing with large numbers or complicated calculations.

Exponentiation

Exponentiation is the mathematical process of raising a number to a power, which is a fundamental operation underpinning the concept of logarithms. When we write \(\b^x\), \(\b\) is the base, and \(\b\) is the exponent. The exponent indicates how many times the base is multiplied by itself. For instance, \(\b^4\) means we multiply \(\b\) by itself four times: \(\b \times \b \times \b \times \b\).

In the context of our exercise, recognizing that \(\b^4 = 625\) is crucial and allows us to deduce that the logarithm \(\bog_5(625) = 4\). Knowing the powers of common bases can significantly speed up the process of simplifying logarithmic expressions. Therefore, gaining a strong grasp of exponentiation is an advantageous skill for working through logarithm-related exercises. Engaging with multiplication tables and practicing squaring and cubing numbers will cultivate an intuitive understanding of exponentiation.

Base of a Logarithm

The base of a logarithm is the number that is being raised to a power. In the given exercise, the base is \(\b\), indicated by \(\bog_5(625)\), meaning we are considering powers of 5. The base plays a pivotal role in any logarithmic expression because the entire calculation depends on the properties of this number.

One important property to remember is that \(\bog_b(1) = 0\) for any base \(\b\), because any number raised to the power of zero equals one. Similarly, \(\bog_b(b) = 1\) since any base raised to the first power is itself. Understanding how the base interacts with different exponents can help simplify the work involved in logarithmic problems. The base also determines the growth rate of the exponential function it's associated with, which is quite relevant in real-world applications like compound interest in finance or population growth in biology.