Solve each exponential equation in Exercises 19-40 by taking...

Solve each exponential equation in Exercises $19-40$ by taking the logarithm on both sides. Express the solution set in terms \mathrm{of logarithms. Then use a calculator to obtain a decimal } ~ approximation, correct to two decimal places, for the solution.
$19^{x}=143$

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Key Concepts

Exponential Equations

Exponential equations are equations where the unknown variable appears in the exponent. These equations require techniques that often involve transforming the equation using logarithms so that the variable can be isolated and solved for.

Logarithms

Logarithms are the inverse functions of exponentials. They are essential in solving exponential equations because they allow us to 'bring down' the exponent, thereby converting an exponential equation into a form that is easier to solve.

Properties of Logarithms

Key properties of logarithms, such as the power rule (which states that log_b(a^c) = c * log_b(a)), are crucial in the process of isolating the variable after taking logarithms on both sides of an equation. This property enables the transformation of an equation where the variable is an exponent into a linear form in terms of the logarithm.

Calculator Approximation

When a solution is expressed in terms of logarithms, a calculator can be used to compute its numerical (decimal) approximation. This step involves evaluating the logarithmic expressions to obtain a practical, approximate value, often rounded to a specified number of decimal places.

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