Eleanor Roosevelt High School
Logarithm WorksheetDefinition For all positive numbers a, where a ≠ 1y = loga x is equivalent to ay = x.A logarithm is an exponent, and loga x is the exponent to which a must be raised to obtain x. The value of x will always be positive and the number a is called the mon logarithmsIf the base is not written, it is assumed to be base 10 and is called a common logarithm. For example log 7 means log10 7.Changing logarithmic equations into exponential equations (and back)You must be able to convert an exponential equation into an equivalent logarithmic equation and vise versa. 32 = 9log39 = 250 = 1___________________________log 1,000 = 3( ? )-1 = 4/3 ___________________________log81 9 = ?Natural logarithmsA number often used as a base for logarithms in many business and scientific applications is the irrational number 2.71817… represented by the letter e. Instead of writing logex, we abbreviate it as ln x. It is read as the natural log of x.e3 = 20.0855…ln 20.0855… = 3e7 = 1096.633…______________________________ln .1353… = -2Using your calculatorYour calculator can find the value for a logarithm if only if it is base 10 or base e (common logs or natural logs). Use your calculator to find the value of each of the following to 3 decimal places:log 1000 = _________ln 16 = _________log 16 = ___________ln e = __________log 348 = __________ln .12 = _________log .25 = __________ln e3 = __________log 1 = ____________ln 1 = ___________Properties of logarithmsFrom the definition of logarithms, you can find that loga 1 = 0loga ak = kalogak = k , (k>0)Rewrite the first 2 of the above equations as an exponential equation and the third one as a logarithmic equation to confirm the property.___________________________________________Since logarithms are exponents and we add exponents when we multiply terms, loga xy = loga x + loga yLikewise loga x/y = loga x - loga y and loga xr = r loga x.See problems on the next page.Change of baseSince your calculator can only find values for base 10 and base e, you must convert other bases into one of these. loga x = logb x= log x = ln xlogb a log aln xFor example log5 17 = log 17 or ln 17 log 5 ln 5Find the value of:log5 17 =log2.9 7.5 = log32.5 =log ? 3 = ................
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