Problem 28 - College Algebra Essentials

Question

Solve each exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$19^{x}=143$$

Step-by-Step Solution

Answer

The solution for the equation \(19^{x}=143\) in terms of a common logarithm is given by \( x = \frac{log(143)}{log(19)}\). After approximating x to two decimal places with a calculator, x should approximately be 2.08.

Step 1: Translate the equation using logarithms

To solve the exponential equation, it can be rewritten using the logarithmic form. The general rule is: if \(a^b = c\), then this is equivalent to \(log_a(c) = b\). So applying this rule to \(19^{x} = 143\) gives \(log_{19}(143) = x \) .

Step 2: Calculate the logarithm

To calculate \(x\), evaluate \(log_{19}(143)\). Most calculators do not directly calculate log base 19, so it will be necessary to use the change of base formula. The change of base formula is \(log_b(a) = \frac{log(a)}{log(b)}\). Applying this rule gives \( x = \frac{log(143)}{log(19)}\).

Step 3: Approximate x

The decimal approximation for the value of x can be obtained with a calculator. Use the calculator to divide the common log of 143 by the common log of 19 to get the approximate value for x, rounding to two decimal places. Note that the 'log' button on most calculators represents the base-10 logarithm.