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Why are negative logarithms undefined?

Explanation

Why are negative logarithms undefined?

Logarithms, written in the form logb(m),\log_b(m), are only defined for m>0.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, log10(100)\log_{10}(100) is the same as asking the question, "What is the exponent needed to raise 1010 to in order to get 100100?" log10(100)=x10x=100 \log_{10}(100)=x \quad \Leftrightarrow \quad 10^x = 100 Here, x=2.x=2. Suppose instead that the value of log10(1) log_{10}(-1) wanted to be found. Exponentially, this can be though of as 10x=-1.10^x=\text{-} 1. The graph of y=10xy=10^x can be used to visualize this equation.

From the diagram, it can be seen that 10x10^x never lies below the xx-axis. This means the function value is never negative. In fact, the function never equals 00 — it gets closer and closer to 0.0. To examine this, let's consider positive and negative values of x.x.

Explanation

Positive values of xx

10x10^x is 1010 multiplied by itself xx times. For example 103=101010. 10^3=10\cdot 10\cdot 10. Since 1010 is a positive number, no matter how many times it is multiplied, the product will always be positive. Therefore, there are no positive xx-values that make 10x10^x negative.

Explanation

Negative xx

Any base raised to a negative exponent can be rewritten as a fraction. x-n=1xn x^\text{-} n = \dfrac{1}{x^n} Can 10x10^x be negative if xx is negative? A base with a negative exponent can be rewritten as a fraction.

xx -1\text{-}1 -2\text{-}2 -3\text{-}3 -4\text{-}4
10x10^x 10-110^{\text{-}1} 10-210^{\text{-}2} 10-310^{\text{-}3} 10-410^{\text{-}4}
Fraction 1101\dfrac{1}{10^1} 1102\dfrac{1}{10^2} 1103\dfrac{1}{10^3} 1104\dfrac{1}{10^4}
== 0.10.1 0.010.01 0.0010.001 0.00010.0001

The "more negative" xx becomes the smaller 10x10^x will be. It gets closer and closer to 00 without actually equaling 00 or becoming negative.

Explanation

Conclusion

Therefore, there are no values of xx that make 10x10^x negative. Since 10x=-110^x=\text{-} 1 has no real solution, log10-1log_{10}\text{-} 1 is undefined.

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