Note that we have exponential expressions on one side of the given biconditional statement. Hence, this is one of the properties of exponential equations.
a^x = a^y
The given exponential expressions are equal to each other and they have the same base. Then, their exponents must also be equal. This is what the One-to-One Property of exponential functions tells us.
a^x = a^y ⇒ x=y
The converse of this statement is also true. If the exponents of two exponential expressions with the same base are equal, then the expressions themselves must be equal, too.
x=y ⇒ a^x = a^y
Let's write the biconditional statement.
a^x = a^y ⇔ x=y
We can now fill in the blank.
a^x = a^y if an only if x=y.
To solve exponential and logarithmic equations, we employ the One-to-One and Inverse Properties of exponential and logarithmic functions. Let's complete the following proposition to represent one of these properties.
log_a x=log_a y if an only if .
Note that we have logarithmic expressions on one side of the given biconditional statement. Hence, this is one of the properties of exponential equations.
log_a x=log_a y
The given logarithmic expressions are equal to each other and they have the same base. Then, their arguments must also be equal. This is what the One-to-One Property of logarithmic functions tells us.
log_a x=log_a y ⇒ x=y
The converse of this statement is also true. If the arguments of two logarithmic expressions with the same base are equal, then the expressions themselves must be equal, too.
x=y ⇒ log_a x=log_a y
Let's write the biconditional statement.
log_a x=log_a y ⇔ x=y
We can now fill in the blank.
log_a x=log_a y if an only if x=y.
Let's complete the following equation to represent one of the One-to-One and Inverse Properties specific to exponential and logarithmic functions.
a^(log_a x)= .
We have an exponential expression with a base a. Its exponent is a logarithmic expression with the same base a. In this case, they undo each other, resulting in the argument x of the logarithmic expression.
a^(log_a x)=x
This is what the Inverse Property of exponential functions tells us. We can now fill in the blank.
a^(log_a x)=x.
Let's complete the following equation to represent one of the One-to-One and Inverse Properties of exponential and logarithmic functions.
log_a a^x= .
We have a logarithmic expression with a base a. Its argument is an exponential expression with the same base a. In this case, they undo each other, resulting in the exponent x of the exponential expression.
log_a a^x=x
This is what the Inverse Property of logarithmic functions tells us. We can now fill in the blank.
