Problem 104 - Algebra and Trigonometry

Question

In Exercises 101–104, write each equation in its equivalent exponential form. Then solve for x. $$ \log _{64} x=\frac{2}{3} $$

Step-by-Step Solution

Answer

The solution to the equation \( \log_{64} x = \frac{2}{3} \) is \( x = 16 \).

Step 1: Converting from Logarithmic to Exponential Form

The equation given is \( \log_{64} x = \frac{2}{3} \), and in the exponential form this becomes \( 64^{\frac{2}{3}} = x \). This follows the general rule of logarithms to exponents: if \( \log_b a = c \), this can be rewritten as \( b^c = a \).

Step 2: Solving for x

Now that the equation is in exponential form, compute the value of \(x\). Solving \( 64^{\frac{2}{3}} \) yields \( x = 16 \). To understand this, remember that \( 64^{\frac{2}{3}} \) can be broken down into two steps. First, compute the cube root of 64 to get 4, then square this to obtain 16.

Key Concepts

Logarithmic to Exponential Conversion

In mathematics, converting a logarithmic equation to its exponential form is a useful skill. This process involves using the rule that if \( \log_b a = c \), then it can be rewritten as \( b^c = a \). For example, consider the equation \( \log_{64} x = \frac{2}{3} \). Here, 64 is the base of the logarithm, \( x \) is the result, and \( \frac{2}{3} \) is the exponent when expressed in exponential form. To convert this into an exponential equation, recognize that it translates literally to: \( x = 64^{\frac{2}{3}} \).Understanding this conversion is crucial because it shifts the problem from the logarithmic domain, which can often seem abstract and difficult to process, into the exponential domain, where direct computation of the exponent is more straightforward.

Solving for x

Once a logarithmic equation is converted into its exponential form, the next step is to solve for \( x \) by evaluating the exponential expression. In our example, the equation \( 64^{\frac{2}{3}} = x \) needs to be solved.**Step-by-Step Calculation:**

• First, find the cube root of 64. The cube root of 64 is 4 because \( 4 \times 4 \times 4 = 64 \).
• Next, take the square of 4 to get the final value. \( 4^2 = 16 \).

So, \( x \) is equal to 16.Breaking down the process into smaller, more manageable steps can simplify the computation and ensure that each step is understood before proceeding to the next.

Logarithmic Equations

Logarithmic equations are equations that involve logarithms with varying bases, and they often pose a challenging aspect of algebra. **Key Points to Remember:**

• Logarithms are the inverse operations of exponentiation. A common task is to rewrite logarithmic expressions as exponential ones, which often results in a simplified equation.
• To solve an equation involving a logarithm, converting from a logarithmic form to an exponential form is typically the first step. This switch can make the solution clearer and more accessible.
• When simplifying a logarithmic equation, consider the properties of exponents. The base and exponent form the foundation of solving the converted equation.

Understanding these properties and techniques is essential as they form the basis for solving not just simple logarithmic equations but more complex problems across algebra and calculus.