# Linking Exponents and Logarithms Examples

### Example 1

Rewrite this logarithm in exponential form: y = log (*x* + 3) | |

The base of this log is 10, so we first write that: 10 The left hand side of the equation is the exponent that goes over the base: 10^{y} The terms inside the log form the other side of the equation: 10^{y} = *x* + 3 | |

### Example 2

Rewrite this exponential in logarithmic form: y = 4(7^{x}) | |

First, we should isolate the *x*: 0.25* y* = 7^{x} Now take the log of both sides, with the base matching the base of the right hand side: The log and base on the right hand side cancel:
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### Example 3

Simplify the following expression: y = 5^{log525} | |

A base raised to a logarithmic exponent is the same as the other way around, so they cancel: *y* = 25 | |

### Example 4

What is the inverse function of *y* = 15*x* + 25? | |

First, we must solve for x. Subtract 25 on both sides: y – 25 = 15x Divide 15: Now switch the variables: | |

### Example 5

What is the inverse function of y = 10^{x} – 4? | |

Add 4 on both sides: *y* + 4 = 10^{x} Take the logarithm of both sides: log(*y *+ 4) = *x* Switch *x* and *y*: *y* = log(*x* + 4) | |

### Example 6

What is the inverse function from the set {1,5 ; 2,8 ; 3,11}? | |

Using the slope-intercept formula, you can recognize this as the function: *y* = 3*x *+ 2 Solve for *x*: *y* – 2 = 3*x* Switch x and y: | |

### Example 7

Simplify: *y* = 9^{log34}
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First, change the base so that it has an exponent the same as the log in the exponent: *y* = 3^{3log34} A base raised to a logarithmic exponent is the same as the other way around, so they cancel: *y* = 3^{4} Solve: *y* = 81
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### Example 8

Does this equation have an inverse:* y* = *x*^{2} + 3*x* + 4? | |

If you graph this function, you will find that it is a parabola This shape fails the horizontal line test, this means that it is not one-to-one Because it is not one-to-one, it cannot have an inverse. | |

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