 
Use this method if the bases can't be changed so they're the same example 1: 5x = 8 log(5x) = log8 Take the log (base 10) of each side xlog5 = log8 Use the power rule. This let's you take the variable out of the exponent x = log8/log5 Solve x = log8/log5 This is the exact answer, as long as it's not evaluated You might see the answer as log58 x = 1.29 (rounded to 2 dp) You can evaluate it if required example 2: 24x + 1 - 3x = 0 We can't just take the log of both sides here, because the log of zero doesn't exist. Instead, rearrange the equation 24x + 1 = 3x We can't make the bases the same, so take the log of both sides log(24x + 1) = log(3x) (4x + 1)log2 = xlog3 Use the power rule. Note the brackets! 4xlog2 + 1log2 = xlog3 Simplify 4xlog2 - xlog3 = - 1log2 Collect the variable terms on one side x(4log2 - log3) = - 1log2 Factor out the variable x = (- 1log2)/(4log2 - log3) Solve This is the exact answer, which can be evaluated if required example 3: 3x + 2 = 7 We can save a step by taking log3 of both sides log33(x + 2) = log37 x + 2 = log37 using the logkkn rule x = log37 - 2 This is the exact answer, which can be evaluated if required x = log7/log3 - 2 or partially evaluated example 4: 5·32x + 4 - 8 = 0 Again, we need to reaarange first because of the zero 5·32x + 4 = 8 We also want just the power on the left side 32x + 4 = 8/5 log332x + 4 = log3(8/5) Take the log3 of both sides 2x + 4 = log3(8/5) Now solve for x 2x = log3(8/5) - 4 x = (log3(1.6) - 4) / 2 This is the exact answer, which can be evaluated if required Or you can partially evaluate it x = (log(1.6)/log3 - 4) / 2 |