Problem 115 - Algebra and Trigonometry

Question

Simplify: \(5^{\log _{5} 19}+\log _{7} 7^{3}\) (Section \(4.2, \text { Example } 5)\)

Step-by-Step Solution

Answer

The simplified result is 22

Step 1: Simplify the first term

Understanding that any base raised to the logarithm of a number, where the logarithm base is the same as the base is the number itself. Like \(b^{\log_{b} x}=x\). So, applying this rule to our first term gives us: \(5^{\log _{5} 19}=19\)

Step 2: Simplify the second term

A logarithm with a base b of the power b^x equals to x. Like \(\log_{b} b^{x} = x\). So, Applying this rule to the second term gives us: \(\log _{7} 7^{3} = 3 \)

Step 3: Add the results

Adding the results obtained from step 1 and step 2 gives us the final answer: \(19+3=22\)

Key Concepts

Understanding Logarithmic Identities

Logarithmic identities are equivalent expressions that involve logarithms. One of the most fundamental identities is that a logarithm of a number to its own base equals that number itself. In mathematical terms, this is expressed as \(\text{{log}}_b(b^x) = x\). This identity is particularly helpful when simplifying logarithmic expressions because it provides a direct method to convert between logarithmic and exponential forms.

To apply this concept to the exercise, remember that \(5^{\log_5 19}\) can be directly simplified to 19, because according to the identity, an exponent and a logarithm with the same base will effectively 'cancel' each other out. Hence, simplifying logarithms often involves looking for such opportunities to apply this identity to make the expression easier to manage.

Properties of Logarithms

Logarithms have unique properties that make them manageable and versatile in mathematical operations. One of the essential properties is the power rule, which states that \(\text{{log}}_b(m^n) = n \times \text{{log}}_b(m)\). This allows us to take an exponent and move it to the front of the logarithm, turning multiplication inside of the logarithm into multiplication outside of it.

For the exercise, this property simplifies the second term, \(\text{{log}}_7(7^3)\). The exponent can come out in front of the log, giving us \(3 \times \text{{log}}_7(7)\), and since the log base is the same as the number, it further simplifies down to 3. The power rule and understanding how to simplify logarithmic expressions using these properties allow for the streamlined solving of complex logarithmic equations.

Exponential and Logarithmic Functions

Exponential and logarithmic functions are inverses of each other. This means that if an exponential function represents continuous growth or decay, the corresponding logarithmic function will represent the time it takes for a certain amount of growth or decay to occur. The inverse nature is marked by the identity \(b^{\log_b x} = x\) for any x > 0 and b > 0, where b ≠ 1.

The exercise provided showcases the inverse relationship beautifully. The first term is an example of an exponential function being simplified by using the inverse logarithmic function. This inverse behavior underscores why the logarithm operation undoes the exponential one, and vice versa, a fundamental concept critical for understanding transformations between exponential and logarithmic forms, which is frequently used in calculus, physics, and engineering.