Use the graph to solve the exponential equation
Use the graph to solve the equation
The graph shows all - points that satisfy the function rule Let's compare the function rule and the equation. The only difference between these two equalities is that the independent variable, is replaced by a in the equation. Thus, we solve the equation by finding the -coordinate of any point on the graph that has the -coordinate
We can identify one such point in the graph. Let's now find the -coordinate of this point graphically.
This -coordinate is not easily read from the graph, so we'll have to make an approximation. It's just a bit bigger than so we'll use This means that an approximate solution to the equation is We can verify this by substituting it into equation to see if a true statement is made.
The right-hand side and the left-hand side are approximately equal, so we have indeed found an approximate solution to the equation:
There are different ways to algebraically solve exponential equations. If both sides of the equation can be written in the same base, equality can be used. For example, consider the equation Since the equation can be written as follows. Now, we have two equivalent expressions with the same base. For the equality to hold, the exponents must also be equal. Thus,
Solve the exponential equation
Solve the equation
When the expressions on both sides of an exponential equation cannot be written with the same base, logarithms can used instead. Essentially, a logarithm is used to undo the exponent. Consider the equation Since is the exponent of base can be applied to each side of the equation. This yields Now, is equal to the exponent needed to make equal This can also be expressed as By using this to rewrite the left-hand side the equation becomes This is the exact solution to the equation. If a numerical value is required, use a calculator. However, the "log" button on a calculator corresponds to a common logarithm — When the logarithm has a base different than the Change of Base rule can be used to express the logarithm with base For the example, equals