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{{ printedBook.courseTrack.name }} {{ printedBook.name }} Logarithms, written in the form $g_{b}(m),$ are only defined for $m>0.$ To prove this, it is necessary to go back to the definition of logarithms. The logarithm of a number is the exponent to which the logarithm's base must be raised in order to get the desired number. For instance, $g_{10}(100)$ is the same as asking the question, "What is the exponent needed to raise $10$ to in order to get $100$?" $g_{10}(100)=x⇔10_{x}=100$ Here, $x=2.$ Suppose instead that the value of $log_{10}(−1)$ wanted to be found. Exponentially, this can be though of as $10_{x}=-1.$ The graph of $y=10_{x}$ can be used to visualize this equation.

From the diagram, it can be seen that $10_{x}$ never lies below the $x$-axis. This means the function value is never negative. In fact, the function never equals $0$ — it gets closer and closer to $0.$ To examine this, let's consider positive and negative values of $x.$

$10_{x}$ is $10$ multiplied by itself $x$ times. For example $10_{3}=10⋅10⋅10.$ Since $10$ is a positive number, no matter how many times it is multiplied, the product will always be positive. Therefore, there are no positive $x$-values that make $10_{x}$ negative.

Any base raised to a negative exponent can be rewritten as a fraction. $x_{-}n=x_{n}1 $ Can $10_{x}$ be negative if $x$ is negative? A base with a negative exponent can be rewritten as a fraction.

$x$ | $-1$ | $-2$ | $-3$ | $-4$ |
---|---|---|---|---|

$10_{x}$ | $10_{-1}$ | $10_{-2}$ | $10_{-3}$ | $10_{-4}$ |

Fraction | $10_{1}1 $ | $10_{2}1 $ | $10_{3}1 $ | $10_{4}1 $ |

$=$ | $0.1$ | $0.01$ | $0.001$ | $0.0001$ |

The "more negative" $x$ becomes the smaller $10_{x}$ will be. It gets closer and closer to $0$ without actually equaling $0$ or becoming negative.

Therefore, there are no values of $x$ that make $10_{x}$ negative. Since $10_{x}=-1$ has no real solution, $g_{10}-1$ is undefined.