Calculate logarithm of a number to any base: Show * Use e for scientific notation. E.g: 5e3, 4e-8, 1.45e12 Logarithm expression calculatorExpression with log(value,base): When: b y = x Then the base b logarithm of a number x: logb x = y Logarithm change of base calculatorAnti-logarithm calculatorIn order to calculate log-1(y) on the calculator, enter the base b (10 is the default value, enter e for e constant), enter the logarithm value y and press the = or calculate button: When y = logb x The anti logarithm (or inverse logarithm) is calculated by raising the base b to the logarithm y: x = logb-1(y) = b y Logarithm rulesLogarithm product rulelogb(x × y) = logb(x) + logb(y) Logarithm quotient rulelogb(x / y) = logb(x) - logb(y) Logarithm power rulelogb(x y) = y × logb(x) Logarithm base switch rulelogb(c) = 1 / logc(b) Logarithm change of base rulelogb(x) = logc(x)/logc(b) Logarithm - log(x) ► See also
Calculator UseThis calculator will solve the basic log equation logbx = y for any one of the variables as long as you enter the other two. The logarithmic equation is solved using the logarithmic function: \( x = \log_{b}b^x \) which is equivalently \( x = b^{log_{b}x} \) How to solve the logarithmic equationIf we have the equation used in the Logarithm Equation Calculator \( \log_{b}x = y \;(1) \) We can say the following is also true \( b^{\log_{b}x} = b^{y} \;(2) \) Using the logarithmic function where \( x = b^{log_{b}x} \) We can rewrite our equation (2) to solve for x \( x = b^{y} \;(3) \) Solving for b in equation (3) we have \( b = \sqrt[y]{x} \) Solving for y in equation (3) \( x = b^{y} \;(3) \) take the log of both sides: \( \log_{10}x = \log_{10}b^y \) Using logarithmic identity we rewrite the equation: \( \log_{10}x = y \cdot \log_{10}b \) Dividing both sides by log b: \( y = \dfrac{\log_{10}x}{\log_{10}b} = \dfrac{\log_{}x}{\log_{}b} \) Note that writing log without the subscript for the base it is assumed to be log base 10 as in log10. Example 1: Solve for y in the following logarithmic equationIf we have \( \log_{3}5 = y \) then it is also true that \( 3^{\log_{3}5} = 3^{y} \) Using the logarithmic function we can rewrite the left side of the equation and we get \( 5 = 3^{y} \) To solve for y, first take the log of both sides: \( \log_{}5 = \log_{}3^y \) By the identity log xy = y · log x we get: \( \log_{}5 = y \cdot \log_{}3 \) Dividing both sides by log 3: \( y = \dfrac{\log_{}5}{\log_{}3} \) Using a calculator we can find that log 5 ≈ 0.69897 and log 3 ≈ 0.4771 2 then our equation becomes: \( n = \dfrac{\log_{}5}{\log_{}3} = \dfrac{0.69897}{0.47712} = 1.46497 \) Therefore, putting y back into our original equation \( \log_{3}5 = 1.46497 \) Example 2: Solve for b in the following logarithmic equationIf we have \( \log_{b}16 = 2 \) then it is also true that \( b^{\log_{b}16} = b^{2} \) Using the logarithmic function we can rewrite the left side of the equation and we get \( 16 = b^{2} \) Solving for b by taking the 2nd root of both sides of the equation \( b = \sqrt[2]{16} = 4 \) Therefore, putting b back into our original equation \( \log_{4}16 = 2 \) Google users came to this page yesterday by typing in these algebra terms:
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