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Chapter 10: Problem 83

Write each logarithmic expression as a single logarithm. $$ k \log 5-\log 4 $$

Short Answer

Expert verified
The logarithmic expression can be simplified as \( \log(5^k/4) \).

Step by step solution

01

Apply the power rule to the first term

Let's start by applying the power rule of logarithms to the first term, this transforms \( k \log 5 \) to \( \log 5^k \). Now our whole term looks like \( \log 5^k - \log 4 \).
02

Apply the quotient rule

The original expression is a difference of two logarithms, which can be written as a single logarithm by using the quotient rule for logarithms. The rule says that the \(\log a - \log b = \log (a/b)\). Applying this, the expression becomes \( \log(5^k / 4) \).

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Key Concepts

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

Power Rule of Logarithms
The power rule of logarithms is a simple but powerful tool. It allows us to move a constant multiple of a logarithm into the exponent of the argument.
For example, if you have an expression like \( k \log b \), you can rewrite it as \( \log(b^k) \). This is extremely helpful when you're trying to simplify logarithmic expressions. By turning the multiplier into an exponent, you reduce the number of terms in your expression.
  • This rule is particularly useful when combining different logarithmic expressions into a single term.
  • It stems from the properties of exponents and their relation to logarithms, since a logarithm is essentially an exponent.
  • Always ensure that the base of the logarithm stays the same when applying this rule.
Understanding this rule allows you to simplify and manage logarithmic expressions more effectively.
Quotient Rule of Logarithms
The quotient rule of logarithms is another handy rule for simplifying expressions. It helps to combine two logarithms that are being subtracted.
If you're faced with \( \log a - \log b \), you can apply this rule to turn it into a single logarithm: \( \log \left( \frac{a}{b} \right) \).
  • This transformation makes the expression more compact and easier to understand.
  • It relies on the idea that logarithms represent powers; subtracting logarithms is akin to dividing powers.
  • Remember that the same base is necessary for this transformation to be valid.
Applying the quotient rule is particularly useful when trying to condense or evaluate expressions without a calculator.
Simplifying Logarithmic Expressions
Simplifying logarithmic expressions involves using rules like the power and quotient rules to streamline messy or complex logarithmic terms.
By using these rules, you can often combine multiple terms into a single, simpler expression. This makes the expression easier to evaluate or further manipulate.
  • Simplification helps in situations where you need a clean form, perhaps for solving equations or comparing values.
  • The key is to apply the rules in a logical order and make sure each step leads to a simpler form.
  • Increasing your familiarity with these rules makes the process second nature and improves your mathematical intuition.
Through practice, simplifying logarithmic expressions becomes a valuable skill for tackling various mathematical problems.

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Most popular questions from this chapter

Multiple choice Which equation represents the ellipse with foci on the \(x\) -axis, major axis 9 units long, minor axis 4 units long, and center at the origin? A \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) B \(\frac{x^{2}}{20.25}+\frac{y^{2}}{4}=1\) C \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) D \(\frac{x^{2}}{4}+\frac{y^{2}}{20.25}=1\)

Explain how to find an equation for the ellipse, centered at the origin, that is 50 units wide and 40 units high.

Find the foci for each equation of an ellipse. $$ 25 x^{2}+4 y^{2}=100 $$

Find the asymptotes of the graph of each equation. $$ y=\frac{5}{x+5}+2 $$

Find the foci for each equation of an ellipse. Then graph the ellipse. $$ 3 x^{2}+y^{2}=9 $$
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