Problem 17 - College Algebra

Question

Write each equation in its equivalent logarithmic form. $$ b^{3}=1000 $$

Step-by-Step Solution

Answer

The equivalent logarithmic form of the given exponential equation \(b^{3}=1000\) is \(log_b 1000 = 3\).

Step 1: Identify Elements in the Exponential Equation

The crucial elements in the equation are: base \(b\), exponent \(3\), and the result \(1000\).

Step 2: Apply the Logarithmic Form

Now change the exponential equation to its equivalent logarithmic form. The base of the exponent becomes the base of the logarithm, and the exponent itself is the result of the logarithm, which is equal to \(1000\). So, the equivalent logarithmic form is \(log_b 1000 = 3\).

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where a variable appears in the exponent. For example, in the equation 5^x = 125, the variable x is the unknown exponent we are trying to find. To solve exponential equations, we often utilize the property that exponential functions are one-to-one, meaning we can apply logarithms to both sides to solve for the variable.

In the exercise provided, we have an exponential equation in the form of b^3 = 1000. To write it in logarithmic form, we take the logarithm of both sides. The base of the exponent becomes the base of our logarithm, and the exponent becomes what the logarithm is equal to. By understanding the inverse relationship between exponentials and logarithms, we can switch from one form to the other to solve for unknowns or to simplify expressions. It's important to recognize the base here, as it will be consistent in both forms of the equation.

Logarithms

Logarithms are the inverses of exponential functions, making them invaluable for solving equations where the unknown is an exponent. The logarithm of a number tells us what exponent we need to raise a base to get that number. Written in mathematical terms, if we have log_b(a) = c, this means that b^c = a.

Using logarithms, we can 'bring down' exponents to where we can handle them - at the level of coefficients. In the context of the exercise b^3 = 1000, by applying a logarithm with base b to both sides (log_b(b^3) = log_b(1000)), we are asking the question, 'to what power must we raise b to obtain 1000?'. This power is 3, thus the logarithmic form is log_b(1000) = 3. Knowledge of logarithms helps with more complex equations, especially when dealing with varied bases or when they are nested within other functions.

Base of an Exponent

The base of an exponent is the number that is being raised to a power. In an equation like 2^5, the number 2 is the base, and it is being raised to the fifth power. Understanding the base is crucial in the context of exponential and logarithmic equations, as it dictates the growth rate of the function and how we approach solving problems involving them.

When converting to logarithmic form, the base of the exponent becomes particularly significant. It's because the logarithm is asking a specific question related to that base: 'to what power do we need to raise this base to achieve a certain number?'. In the exercise b^3 = 1000, the base b is the number we are raising to the third power to get 1000, and when we apply a logarithm, it remains the base in the logarithmic expression log_b(1000). Recognizing and working with the base is essential for correctly translating between exponential and logarithmic forms.