Solve each exponential equation by taking the logarithm on...

Solve each exponential equation by taking the logarithm on both sides. Express the solution set in terms of logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution.
$$9 e^{x}=99$$

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Key Concepts

Properties of Logarithms

The properties of logarithms, such as the product, quotient, and power rules, are essential when solving exponential equations. These rules allow for the simplification of logarithmic expressions, for example, turning the logarithm of a product into a sum of logarithms, which aids in isolating the variable to solve the equation.

Calculator Approximations

Once the exact solution is obtained in terms of logarithms, numerical approximation using a calculator enables the expression of the solution in decimal form. This is particularly useful for practical applications where a rounded numerical value is more informative than an exact logarithmic expression.

Exponential Equations

An exponential equation is one in which the variable appears in the exponent of a constant base. Solving these equations often requires techniques that transform the exponential form into a linear one, so that standard algebraic methods can be applied to isolate the variable.

Logarithms

Logarithms serve as the inverse operations to exponentiation, allowing one to isolate the variable in the exponent by converting the exponential equation into a form that is easier to solve. They provide the necessary tool to handle equations where the unknown is in the exponent.

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