Problem 22 - Precalculus

Question

Write each exponential equation in its equivalent logarithmic form. $$3^{6}=729$$

Step-by-Step Solution

Answer

The logarithmic form is \( 6 = \log_3 729 \).

Step 1: Identify Exponential Form

The given exponential equation is \( 3^6 = 729 \), where 3 is the base, 6 is the exponent, and 729 is the result.

Step 2: Understand Logarithmic Form

The characteristic of logarithms is that they allow us to express the exponent as a function of the base and the result. In logarithmic terms, if \( a^b = c \), then this can also be written as \( b = \log_a c \).

Step 3: Apply Logarithmic Conversion

Using the relation from Step 2, replace the known values into the logarithmic form. For the equation \( 3^6 = 729 \), this means that 6 is the logarithm power, 3 is the base, and 729 is the result, so it becomes: \( 6 = \log_3 729 \).

Step 4: Write Final Logarithmic Equation

The equivalent logarithmic form of the equation \( 3^6 = 729 \) is \( 6 = \log_3 729 \), which means that 6 is the power to which 3 must be raised to get 729.

Key Concepts

Exponential Equations

An exponential equation is a mathematical expression where a constant base is raised to a power or exponent to get a particular result. In our example, the exponential equation is \(3^6 = 729\). Here, 3 is the base, and 6 is the exponent. Exponential equations are useful in describing phenomena that grow or decay at consistent rates, such as compound interest or population growth.
When solving exponential equations, especially those where you need to find an unknown exponent or result, logarithms are a fantastic tool. They help us "unwrap" the exponent, making the equations easier to solve. This process involves converting the exponential form into its equivalent logarithmic form.

Logarithms

Logarithms are the inverse operations of exponentials. They answer the question: what exponent do we need to raise a specific base to, in order to obtain a given number? For instance, with the logarithmic form \( 6 = \log_3 729 \), you can easily check it in reverse: raising 3 to the 6th power gives us 729.
The general logarithmic form is \( b = \log_a c \), where \(a^b = c\). This expression reads as: **b is the power** to which the base **a** must be raised to get **c**. Logarithms are particularly handy in solving equations where the unknown is in the exponent, making them incredibly valuable in fields ranging from science to finance.

Base and Exponent

Within exponential equations and logarithms, the terms 'base' and 'exponent' are fundamental. **The base** is the number being multiplied by itself, and it determines the factor repeated multiplication. In \(3^6 = 729\), 3 is the base. Changing the base affects the entire equation dramatically. For example, \(2^6\) is 64, not 729.
**The exponent**, often referred to as the power, shows how many times the base is multiplied by itself. In our exercise, the exponent is 6, indicating that the base 3 is multiplied by itself 6 times. Understanding these terms is crucial when working with any exponential or logarithmic form, as they define the relationship in these equations. Moreover, recognizing how to manipulate these components helps in converting between exponential and logarithmic forms effectively.