If you need help solving logarithms, Arithmetic.com is a helpful, free resource that provides information regarding basic mathematical concepts, including the three laws of logarithms. The three laws of logarithms provide an algorithm based on three simple formulas that can help you immensely when solving logarithms. In order to understand these concepts, you must first understand that a logarithm is the exponent by which a specific number would have to be raised in order to produce the number for which you are solving, x. In this way, logarithms serve as the inverse function of exponents. As we shall define more clearly a bit later on, the logarithm of a particular number, x, raised to a particular power, n, is equal to the logarithm of x (the exponent) times n (the coefficient). The first law of logarithms is known as multiplication becomes addition. This means that the logarithm of a given product is equal to the sums of the logarithms of each of the product’s base components. Put into formulaic terms, the logarithm of xy is equal to the logarithm of x plus the logarithm of y. The second law of logarithms is known as division becomes subtraction. This means that the logarithm of a given quotient is equal to the difference of the logarithms of each of the quotient’s base components. The logarithm of x divided by y is equal to the difference obtained by subtracting the logarithm of y from the logarithm of x. The final law of logarithms is known as exponent becomes multiplier. In this case, the number obtained by finding the logarithm of x raised to a specific power (we will call this power “n”) is equal to multiplying the logarithm of x by that number, n. Note that in this formula, we are not solving for the entire logarithm being raised to a power n, but rather we are finding the logarithm of just the x being raised to n. Conversely, we can find this number by multiplying the entirety of the logarithm by n used as a coefficient.
The three laws of logarithms can also be effectively combined when solving logarithms. This can best be demonstrated by use of example. To find the logarithm of ab divided by c, you would combine the rules of multiplication and division. The result therefore would be the logarithm of a plus the logarithm of b minus the logarithm of c. In a slightly more complicated example, we can find the logarithm of a times b squared all divided by c to the fourth power. In this case, the result would be equal to the logarithm of a plus two times the logarithm of b (making the exponent the coefficient) minus four times the logarithm of c. For more examples of solving logarithms using the three laws of logarithms, check out further articles on Arithmetic.com.