Problem 104 - College Algebra

Question

Write each equation in its equivalent exponential form. Then solve for \(x .\) $$ \log _{64} x=\frac{2}{3} $$

Step-by-Step Solution

Answer

The equivalent exponential form of the equation is \(64^{\frac{2}{3}}=x\) and the solution for \(x\) is 16.

Step 1: Convert the logarithmic equation into an exponential equation

We use the formula \(b^{y}=a \) for the notation \( \log_{b} a=y .\) Substituting the given values we get \(64^{\frac{2}{3}}=x.\)

Step 2: Simplify the base to solve for \(x\)

We rewrite 64 as \(2^6\) and utilize the property that \((a^{b})^{c}=a^{bc}\) simplifying the equation to \( (2^6)^{\frac{2}{3}}=x,\) resulting in \(2^4=x.\)

Step 3: Calculate the value of \(x\)

We calculate the value of \(2^4\) to get \(x=16.\)

Key Concepts

Exponential Form

Understanding the exponential form is critical for transforming a logarithmic equation and solving it effectively. The exponential form relates to expressing a number as a base raised to a power. For example, the number 8 can be expressed as the exponential form of 2^3 because the base 2 raised to the power of 3 equals 8.

In the case of the exercise provided, 64^(2/3)=x is an exponential form where 64 is the base, and 2/3 is the exponent. This expression indicates that we are looking for the cube root of 64 squared, as raising a number to the power of 1/3 is the same as taking the cube root. Understanding this concept allows us to simplify the equation and find the value of x.

Solving Exponential Equations

Solving exponential equations often requires altering the exponent or the base to make calculations easier. An important strategy in solving these equations is to express the base as a power of a common base when possible.

Let's dissect the steps taken in the solution. Starting with 64^(2/3)=x, we recognize that 64 is 2^6. Utilizing the property that (a^b)^c = a^(bc), we're able to simplify 64^(2/3) to 2^(6*(2/3)), which further simplifies to 2^4. Finally, calculating the value of 2^4 gets us to the solution x=16. Learning to recognize and use exponent properties will make solving these types of equations much more manageable.

Logarithm Properties

The properties of logarithms are essential tools for solving logarithmic equations. One of these properties is the inverse relationship between logarithms and exponents, which states that b^y=a is equivalent to log_b(a)=y. This relationship allows you to convert between logarithmic and exponential forms, which is a critical step in solving logarithmic equations.

Other important logarithm properties include the product rule, quotient rule, and power rule, which provide methods to simplify complex logarithmic expressions. By understanding these properties, you can break down and solve logarithmic equations that, at first glance, might seem daunting. The exercise demonstrates this by converting a logarithm to its exponential form, an application of the inverse nature of logs and exponents.