Problem 55 - Precalculus with Calculus Previews

Question

In Problems \(51-56\), use the laws of logarithms in Theorem \(6.2 .1\) to rewrite the given expression as one logarithm. $$ \ln 5+\ln 5^{2}+\ln 5^{3}-\ln 5^{6} $$

Step-by-Step Solution

Answer

The expression simplifies to 0.

Step 1: Apply the Power Rule for Logarithms

According to the power rule: \( \ln(a^b) = b\ln(a) \). We will apply this rule to each term in the expression where a power is present. Therefore, \( \ln(5^2) = 2\ln(5) \) and \( \ln(5^3) = 3\ln(5) \). Now the expression becomes \( \ln(5) + 2\ln(5) + 3\ln(5) - \ln(5^6) \).

Step 2: Combine Like Terms

We have multiple \( \ln(5) \) terms. Combine these using arithmetic addition: \( \ln(5) + 2\ln(5) + 3\ln(5) = (1+2+3)\ln(5) = 6\ln(5) \). Now, the expression simplifies to \( 6\ln(5) - \ln(5^6) \).

Step 3: Apply the Power Rule Again

Apply the power rule to \( \ln(5^6) \) to rewrite it as \( 6\ln(5) \). The current form of the expression is \( 6\ln(5) - 6\ln(5) \).

Step 4: Simplify the Expression

Use combining like terms again. Since \( 6\ln(5) - 6\ln(5) \) equals zero, the logarithmic expression simplifies to zero.

Key Concepts

Power Rule for Logarithms

The power rule is an exciting tool when working with logarithms. It allows you to bring down the exponent in a logarithmic expression as a multiplier. This means that if you have an expression like \( \ln(a^b) \), it can be rewritten as \( b \ln(a) \).
Here's why the power rule is so valuable:

• It simplifies expressions by turning powers into simple multiples.
• It makes complex logarithmic expressions easier to handle and manipulate.

Consider our exercise, where you saw terms like \( \ln(5^2) \) and \( \ln(5^3) \). Using the power rule, these terms were transformed to \( 2\ln(5) \) and \( 3\ln(5) \) respectively.
This change is crucial because working with simple multiples is far easier than dealing with powers directly.
Always remember to check for exponents in log expressions and see if the power rule can be applied, making your calculations a breeze.

Combining Logarithms

Combining logarithms is another powerful technique to simplify expressions. Once you've applied the power rule, you often find terms that are similar and can be merged.
In our example expression, we have the summed terms \( \ln(5) + 2\ln(5) + 3\ln(5) \). By combining these, we add the coefficients of \( \ln(5) \), resulting in \( 6\ln(5) \).

• Combine like terms to condense the expression.
• This step helps simplify and solve the problem efficiently.

Once combined, it's easier to see the next steps, which may further simplify the expression. Just like with regular numbers, by adding logarithms that involve similar terms, you create a more manageable equation.

Natural Logarithm

Natural logarithms, denoted as \( \ln \), are logarithms with the base \( e \), where \( e \approx 2.718 \). These are widely used in mathematics, especially in calculus, because of their unique properties related to exponential functions.
The symbol \( \ln \) is crucial because it sets the context of the base being \( e \), which is special in math for its naturally occurring nature.

• \( e \) is an irrational number, making \( \ln \) distinct from other logarithms like base 10 or base 2.
• \( \ln \) simplifies many equations, especially those involving growth and decay processes.

In solving the problem, using \( \ln \) allowed the application of specific rules apt for the natural logarithm. It's vital to recognize the base when working logarithms, helping to employ the appropriate strategies and rules.