neve domain was triggered too early. This is usually an indicator for some code in the plugin or theme running too early. Translations should be loaded at the init action or later. Please see Debugging in WordPress for more information. (This message was added in version 6.7.0.) in /home/mathze5/public_html/learnermath.com/wp-includes/functions.php on line 6114Logarithms and examples of working with logarithms were introduced on the logarithms<\/i><\/a> page, and the exponential and logarithmic functions examples<\/i><\/a> page.\n When learning how to simplify logarithms, there is a formula that can be very helpful when we wish to establish the value of a given logarithm.\n
\nHere on this how to simplify logarithms page we look further at situations of manipulating and solving where logarithms are present.\n
\nIncluding the logarithms change of base formula.\n
\nThere are some rules of logarithms that can be used in certain examples where we want to express or solve for an unknown. These rules actually follow on from the exponent rules seen on the exponents in algebra<\/i><\/a> page.\n
\nWe can firstly recap the key rules for exponents.<\/p>\n
\n\n\n\nExponent Rules:<\/span><\/h2>\n\n\n\n
\n–<\/b><\/font> a^m \\times a^n \\space {\\small{=}} \\space a^{m + n}<\/span><\/font>\n
\n–<\/b><\/font> {\\Large{\\frac{a^m}{a^n}}} \\space {\\small{=}} \\space a^{m {\\text{--}} n}<\/span><\/font>\n
\n–<\/b><\/font> (a^m)^n \\space {\\small{=}} \\space a^{mn}<\/span><\/font>\n
\n–<\/b><\/font> a^0 \\space {\\small{=}} \\space 1<\/span><\/font> , a^1 \\space {\\small{=}} \\space a<\/span><\/font>\n
\nThe rules for logarithms follow in a similar pattern.\n
\n<\/a>\n
\n\n\n\nLogarithm Rules:<\/span><\/h2>\n\n\n\n
\n–<\/b><\/font> {\\tt{log}}_a (mn) \\space {\\small{=}} \\space {\\tt{log}}_a (m) + {\\tt{log}}_a (n)<\/span><\/font>\n
\n–<\/b><\/font> {\\tt{log}}_a (\\frac{m}{n}) \\space {\\small{=}} \\space {\\tt{log}}_a (m) \\space {\\text{--}} \\space {\\tt{log}}_a (n)<\/span><\/font>\n
\n–<\/b><\/font> {\\tt{log}}_a (x^n) \\space {\\small{=}} \\space n \\space {\\tt{log}}_a (x)<\/span><\/font>\n
\n–<\/b><\/font> {\\tt{log}}_a (a) \\space {\\small{=}} \\space 1<\/span><\/font> , {\\tt{log}}_a (1) \\space {\\small{=}} \\space 0<\/span><\/font>\n
\n<\/a>\n
\n\n\n\n
\n\n\n\nLogarithm Rules Examples<\/span><\/h2>\n\n\n\n
\n(1.1) <\/i><\/b><\/font><\/u><\/font><\/font>\n
\na) <\/i><\/b><\/font><\/font><\/font><\/u> Write {\\tt{log}} (2a)<\/span><\/font> as the sum of two logs.\n
\nSolution<\/i> <\/b><\/font><\/font><\/u><\/font>\n
\n{\\tt{log}} (2a) \\space {\\small{=}} \\space {\\tt{log}}(2) + {\\tt{log}}(a)<\/span><\/font>\n
\nb) <\/i><\/b><\/font><\/font><\/font><\/u> Write {\\tt{log}}b^4<\/span><\/font> as a multiple.\n
\nSolution<\/i> <\/b><\/font><\/font><\/u><\/font>\n
\n{\\tt{log}}b^4 \\space {\\small{=}} \\space \\space 4\\space{\\tt{log}}b<\/span><\/font>\n
\n(1.2) <\/i><\/b><\/font><\/u><\/font><\/font>\n
\nWe will show all of the following written as the log of a single number.\n
\na) <\/i><\/b><\/font><\/font><\/font><\/u> {\\tt{log}} (8) \\space {\\text{--}} \\space {\\tt{log}} (4)<\/span><\/font> => {\\tt{log}} (8) \\space {\\text{--}} \\space {\\tt{log}} (4) \\space {\\small{=}} \\space {\\tt{log}} (\\frac{8}{4}) \\space {\\small{=}} \\space {\\tt{log}} (2)<\/span><\/font>\n
\nb) <\/i><\/b><\/font><\/font><\/font><\/u> 3{\\tt{log}}4<\/span><\/font> => 3{\\tt{log}}4 \\space {\\small{=}} \\space {\\tt{log}}4^3 \\space {\\small{=}} \\space {\\tt{log}}64<\/span><\/font>\n
\nc) <\/i><\/b><\/font><\/font><\/font><\/u> {\\text{-}}2{\\tt{log}}5<\/span><\/font> => {\\text{-}}2{\\tt{log}}5 \\space {\\small{=}} \\space {\\tt{log}}5^{{\\text{-}}2} \\space {\\small{=}} \\space {\\tt{log}} {{\\Large{\\frac{1}{5^2}}}} \\space {\\small{=}} \\space {\\tt{log}} {{\\Large{\\frac{1}{25}}}}<\/span><\/font>\n
\nd) <\/i><\/b><\/font><\/font><\/font><\/u> 2{\\tt{log}}2 \\space {\\text{--}} \\space 3{\\tt{log}}3<\/span><\/font> => 2{\\tt{log}}2 \\space {\\text{--}} \\space 3{\\tt{log}}3 \\space {\\small{=}} \\space {\\tt{log}}2^2 \\space {\\text{--}} \\space {\\tt{log}}3^3<\/span> \\space {\\small{=}} \\space {\\tt{log}}4 \\space {\\text{--}} \\space {\\tt{log}}27 \\space {\\small{=}} \\space {\\tt{log}} \\frac{4}{27}<\/span><\/font>\n
\n(1.3) <\/i><\/b><\/font><\/u><\/font><\/font>\n
\nLike in examples (1.2), we will show all of the following written as the log of a single number.\n
\na) <\/i><\/b><\/font><\/font><\/font><\/u> 1 + {\\tt{log}}_3 (2)<\/span><\/font> => 1 + {\\tt{log}}_3 (2) \\space {\\small{=}} \\space {\\tt{log}}_3 (3) + {\\tt{log}}_3 (2) \\space {\\small{=}} \\space {\\tt{log}}_3 (2 \\times 3) \\space {\\small{=}} \\space {\\tt{log}}_3(6)<\/span><\/font>\n
\nb) <\/i><\/b><\/font><\/font><\/font><\/u> 2 + {\\tt{log}}_{10} (5)<\/span><\/font> => 2 + {\\tt{log}}_{10} (5) \\space {\\small{=}} \\space 2{\\tt{log}}_{10} (10) + {\\tt{log}}_{10} (5) \\space {\\small{=}} \\space {\\tt{log}}_{10}(10^2) + {\\tt{log}}_{10}(5)<\/span><\/font>\n
\n {\\small{=}} \\space {\\tt{log}}_{10} (100) + {\\tt{log}}_{10} (5) \\space {\\small{=}} \\space {\\tt{log}}_{10} (100 \\times 5) \\space {\\small{=}} \\space {\\tt{log}}_{10}(500)<\/span><\/font>\n
\nc) <\/i><\/b><\/font><\/font><\/font><\/u> 1 + {\\tt{log}}_5 b<\/span><\/font> => 1 + {\\tt{log}}_5 b \\space {\\small{=}} \\space {\\tt{log}}_5 5 + {\\tt{log}}_5 b \\space {\\small{=}} \\space {\\tt{log}}_b (5 \\times b) \\space {\\small{=}} \\space {\\tt{log}}_5 5b<\/span><\/font>\n
\n(1.4) <\/i><\/b><\/font><\/u><\/font><\/font>\n
\nExpress {\\tt{log}}_{10} (20)<\/span><\/font> in terms of {\\tt{log}}_{10} (2)<\/span><\/font> and {\\tt{log}}_{10} (5)<\/span><\/font>.\n
\nSolution<\/i> <\/b><\/font><\/font><\/u><\/font>\n
\n{\\tt{log}}_{10} (20) \\space {\\small{=}} \\space {\\tt{log}}_{10} (4 \\times 5) \\space {\\small{=}} \\space {\\tt{log}}_{10} (4) + {\\tt{log}}_{10} (5)<\/span><\/font>\n
\n{\\small{=}} \\space {\\tt{log}}_{10} (2)^2 + {\\tt{log}}_{10} (5) \\space {\\small{=}} \\space 2{\\tt{log}}_{10} (2) + {\\tt{log}}_{10} (5)<\/span><\/font>\n
\n(1.5) <\/i><\/b><\/font><\/u><\/font><\/font>\n
\nSolve for x<\/span>: {\\tt{log}}x + 2{\\tt{log}}2 \\space {\\small{=}} \\space {\\tt{log}}48<\/span><\/font>.\n
\nSolution<\/i> <\/b><\/font><\/font><\/u><\/font>\n
\n{\\tt{log}}x + 2{\\tt{log}}2 \\space {\\small{=}} \\space {\\tt{log}}48<\/span><\/font> => {\\tt{log}}x + {\\tt{log}}2^2 \\space {\\small{=}} \\space {\\tt{log}}48<\/span><\/font> => {\\tt{log}}x + {\\tt{log}}4 \\space {\\small{=}} \\space {\\tt{log}}48<\/span><\/font>\n
\n=> {\\tt{log}}4x \\space {\\small{=}} \\space {\\tt{log}}48<\/span><\/font> , 4x \\space {\\small{=}} \\space 48 \\space \\space , \\space \\space x \\space {\\small{=}} \\space 12<\/span><\/font>\n
\n
\n<\/a>\n
\n\n\n\n
\n\n\n\nLogarithms Change of Base Formula<\/span><\/h2>\n\n\n\n
\n
\nIt is called the logarithms change of base formula.<\/p>\n
\nChange of Base Formula => {\\tt{log}}_a b \\space {\\small{=}} \\space {\\Large{\\frac{{\\tt{log}}_c b}{{\\tt{log}}_c a}}}<\/span><\/font>\n
\nThe value of c<\/i><\/font> can be any value we wish, and we can see in an example why this is helpful.\n
\nExample <\/font><\/font><\/font><\/u><\/b>\n
\n(2.1) <\/i><\/b><\/font><\/font><\/font><\/u>\n
\nEstablish the value of {\\tt{log}}_4 9<\/span><\/font> to 4 decimal places.\n
\nSolution<\/i> <\/b><\/font><\/font><\/u><\/font>\n