Q&A for How to Solve Logarithms

Logarithmic graphs have asymptotes at x=0, therefore any negative answers cannot exist.

Understand that the antilog of 12.18 is a common antilogarithm question, which means the assumed subscript is 10. To solve this problem, take 10 to the power of 12.18. This gives you an approximate value of 1,513,561,248,436.21.

4^x + 10 = 10. Subtract 10 from both sides. 4^x = 0. Rewrite according to the definition of a logarithm. log4(0) = x. Use the change of base formula to calculate log4(0). log(0)/log(4) = x log(0) is undefined, making x undefined as well.

Since this equation hasn't been set equal to anything and has no x's to solve for, we can solve by simply plugging it into a calculator, giving us 5.799025... as our answer.

First, simplify the left-hand side. As a rule, log(a) - log(b) = log(a/b), so we rewrite as: log10[(x+2) / (x-1)] = 2. Next we need to convert the right-hand side to a logarithm so we can cancel them out on both sides. Because loga(a) = 1 (log of a to base a), log10[(x+2) / (x-1)] = 2*log10(10). The right-hand side MUST then be written as log10(10^2) BEFORE attempting to cancel out the logs, since nlog(a) = log(a^n). Cancelling out the logarithms from both sides leaves us with: (x+2) / (x-1) = 100. From here, we simply solve for x using basic algebra: x+2 = 100(x-1) x+2 = 100x - 100 100x - x = 2 + 100 99x = 102 x = 102/99.

Try to find 168.75 in terms of 16 and 81. After trying to divide by 16 and 81, you can determine that 25x81/12 = 168.75. Since we have the term 81 as a factor, we have to try to find 16 as a factor. We can find the factors of 16 as 4x4, 2x2x2x2 and the factors of 12 as 3x4. We can neglect 25, because it is 5x5 and there are no common factors with 81 and 16. Multiply top and bottom by 4 to get 4x25x81/(3x4x4) = 100x81/(3x16). Now that we have our factors, we can express it in terms of m and n by taking logarithms. Log(100x81/(3x16)) = log(100) + log(81) - (log(3) + log(16)). Using properties of log, log(81)=log(9²)=2log(9)=2log(3²)=4log(3)=n, so log(3)=n/4. Log(100) is log(10²) or 2log(10)=2x1. By substitution, log(100x81/(3x16))=2+n-n/4-m=2+3n/4+m.

3log10 (3)=log10 (1/x) => log10 (3^3)=log10 (1/x) or 3^3=1/x, which gives x=1/(3^3) or 1/27

Find log2(32). You get 5. So log2(x + 2) - 2 = 5 and log2(x + 2) = 7. Then put 2 as the base for both sides: x + 2 = 2⁷ = 128, so that x = 128 - 2 = 126.

You would have to use a calculator to find the exact value.

Combine these two into log 10 ((10 + 9x)/(11 - x)) = 2. Then (10 + 9x )/(11 - x) = 100. 10 + 9x = 100(11 - x). 10 + 9x = 1100 - 100x. Add 100x to both sides, and subtract 10 to both sides. 109x = 1090. x = 10.

The base is the same on both sides, so this means that (7x + 3) = (5x + 9). Or 7x + 3 = 5x + 9. Subtract 5x on both sides: 2x + 3 = 9, and subtract 3 on both sides: 2x = 6, therefore x = 3.

Using log(a)+log(b) = log(ab), we find that log(x+1) = log(3x). Then that means x + 1 = 3x. Then to finish, subtract x on both sides. 1 = 2x. Then x = 1/2.

Exponentiate both sides of the equation by making the log base the same on both sides. The base and log will cancel, and you have an exponential equation. Similarly, to change an exponential equation to a log one, take the log of the same base on both sides, the base and log will cancel, and you get a logarithmic equation.

This would be hard to do by hand, so you would have to use a calculator to find the exact value, which most calculators read as 2.890812098955124.

Divide 85.32 by 9.82, then square that answer. You'll get an approximate answer of 75.49.

Log 3 x = 2 means that x = 3². So x = 9. (The way to say "log 3 x = 2" is "the log to the base 3 of x is 2." Having solved for x, we can say "the log to the base 3 of 9 is 2.")