/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 Simplify. $$ \log _{5} 625 ... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 19

Simplify. $$ \log _{5} 625 $$

Short Answer

Expert verified
4

Step by step solution

01

- Identify the base

The given expression is \(\text{log}_{5} 625\). The base here is 5.
02

- Express 625 as a power of the base

Rewrite 625 as a power of 5. Since \[ 5^4 = 625 \], we have \[ 625 = 5^4 \].
03

- Apply the logarithm rule

Use the logarithm power rule \[ \text{log}_b(b^x) = x \]. Applying this to the expression, we get \[ \text{log}_{5}(5^4) = 4 \].
04

- State the simplified result

Thus, the simplified form of \(\text{log}_{5} 625\) is \(\boxed{4}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

logarithm base
A logarithm answers the question: 'To what exponent must we raise a particular number (the base) to get another number?' Think of it as the inverse operation of exponentiation. In \ \(\text{log}_{5} 625\ \), the base is 5. This means we are trying to find the power or exponent to which 5 must be raised to yield 625. A helpful tip is to always identify the base in any logarithmic expression. It's usually the small number written just below and to the right of the log symbol.
logarithm power rule
The logarithm power rule is very useful in simplifying expressions involving logarithms. The rule states: \[ \text{log}_b(b^x) = x \]. This means if you have a logarithm of a base that has been raised to a power, the solution is simply the exponent. For example, when we simplified \(\text{log}_{5}(5^4)\), we applied this rule and found that it simplifies to 4. This rule works because logarithms are the inverse of exponentiation. Their roles in math are to undo each other, making complex expressions more manageable.
exponents
Exponents tell us how many times to multiply a number by itself. For example, \[ 5^4 = 5 \times 5 \times 5 \times 5 = 625 \]. In our exercise, you saw how we rewrote 625 as \(5^4\). Recognizing patterns in numbers and expressing them as powers of a common base can greatly simplify logarithmic expressions. This is why understanding exponents is crucial for mastering logarithms. Whenever you face a logarithmic problem, try expressing the number as a power of the base. This can often turn a complicated problem into a straightforward one.

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Most popular questions from this chapter

Express as an equivalent expression that is a single logarithm and, if possible, simplify. $$\frac{1}{2} \log _{a} x+5 \log _{a} y-2 \log _{a} x$$

$$\text { If } \log _{a} x=2, \text { what is } \log _{1 / a} x ?$$

Express as an equivalent expression that is a single logarithm and, if possible, simplify. $$\log _{a} x^{2}-2 \log _{a} \sqrt{x}$$

Use the properties of logarithms to find each of the following. $$\log _{5}(125 \cdot 625)$$

Solve for \(x\) $$ \log (492 x)=5.728 $$
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