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Explanation

# Why are negative logarithms undefined?

Logarithms, written in the form $\log_b(m),$ 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, $\log_{10}(100)$ is the same as asking the question, "What is the exponent needed to raise $10$ to in order to get $100$?" $\log_{10}(100)=x \quad \Leftrightarrow \quad 10^x = 100$ Here, $x=2.$ Suppose instead that the value of $log_{10}(-1)$ wanted to be found. Exponentially, this can be though of as $10^x=\text{-} 1.$ 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 $x.$

Explanation

## Positive values of $x$

$10^x$ is $10$ multiplied by itself $x$ times. For example $10^3=10\cdot 10\cdot 10.$ 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.

Explanation

## Negative $x$

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

$x$ $10^x$ Fraction $=$ $\text{-}1$ $\text{-}2$ $\text{-}3$ $\text{-}4$ $10^{\text{-}1}$ $10^{\text{-}2}$ $10^{\text{-}3}$ $10^{\text{-}4}$ $\dfrac{1}{10^1}$ $\dfrac{1}{10^2}$ $\dfrac{1}{10^3}$ $\dfrac{1}{10^4}$ $0.1$ $0.01$ $0.001$ $0.0001$

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.

Explanation

## Conclusion

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