<< Chapter < Page | Chapter >> Page > |
Is there any way to solve
Yes. The solution is
One common type of exponential equations are those with base This constant occurs again and again in nature, in mathematics, in science, in engineering, and in finance. When we have an equation with a base on either side, we can use the natural logarithm to solve it.
Given an equation of the form solve for
Does every equation of the form have a solution?
No. There is a solution when and when and are either both 0 or neither 0, and they have the same sign. An example of an equation with this form that has no solution is
Solve
Sometimes the methods used to solve an equation introduce an extraneous solution , which is a solution that is correct algebraically but does not satisfy the conditions of the original equation. One such situation arises in solving when the logarithm is taken on both sides of the equation. In such cases, remember that the argument of the logarithm must be positive. If the number we are evaluating in a logarithm function is negative, there is no output.
Does every logarithmic equation have a solution?
No. Keep in mind that we can only apply the logarithm to a positive number. Always check for extraneous solutions.
We have already seen that every logarithmic equation is equivalent to the exponential equation We can use this fact, along with the rules of logarithms, to solve logarithmic equations where the argument is an algebraic expression.
For example, consider the equation To solve this equation, we can use rules of logarithms to rewrite the left side in compact form and then apply the definition of logs to solve for
Notification Switch
Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?