For example, 65+y=63{\displaystyle 6^{5+y}=6^{3}} has an exponent on either side of the equation, and each exponent has the same base (6).
For example, in the equation 65+y=63{\displaystyle 6^{5+y}=6^{3}}, since both exponents have the same base, you would write an equation for the exponents: 5+y=3{\displaystyle 5+y=3}.
For example:5+y=3{\displaystyle 5+y=3}5+y−5=3−5{\displaystyle 5+y-5=3-5}y=−2{\displaystyle y=-2}
For example, if you found that y=−2{\displaystyle y=-2}, you would substitute −2{\displaystyle -2} for y{\displaystyle y} in the original equation:65+y=63{\displaystyle 6^{5+y}=6^{3}}65−2=63{\displaystyle 6^{5-2}=6^{3}}63=63{\displaystyle 6^{3}=6^{3}}
For example, if your are trying to solve 3x−5−2=79{\displaystyle 3^{x-5}-2=79}, you first need to isolate 3x−5{\displaystyle 3^{x-5}} by adding 2 to each side of the equation:3x−5−2=79{\displaystyle 3^{x-5}-2=79}3x−5−2+2=79+2{\displaystyle 3^{x-5}-2+2=79+2}3x−5=81{\displaystyle 3^{x-5}=81}
For example, look at the equation 3x−5=81{\displaystyle 3^{x-5}=81}. You need to change 81 to an exponent with a base of 3, so that it matches the other exponential expression in the equation. By factoring out 3, you should see that 3×3×3×3=81{\displaystyle 3\times 3\times 3\times 3=81}, so 34=81{\displaystyle 3^{4}=81}. The new equation then becomes 3x−5=34{\displaystyle 3^{x-5}=3^{4}}.
For example, since 3x−5=34{\displaystyle 3^{x-5}=3^{4}} has two exponents with a base of 3, you can ignore the base and simply look at the equation x−5=4{\displaystyle x-5=4}.
For example:x−5=4{\displaystyle x-5=4}x−5+5=4+5{\displaystyle x-5+5=4+5}x=9{\displaystyle x=9}
For example, if you found that x=9{\displaystyle x=9}, you would plug in 9{\displaystyle 9} for x{\displaystyle x} in the original equation and simplify:3x−5=81{\displaystyle 3^{x-5}=81}39−5=81{\displaystyle 3^{9-5}=81}34=81{\displaystyle 3^{4}=81}81=81{\displaystyle 81=81}
For example, you need to isolate the expression 43+x{\displaystyle 4^{3+x}} in the equation 43+x−8=17{\displaystyle 4^{3+x}-8=17} by adding 8 to both sides:43+x−8=17{\displaystyle 4^{3+x}-8=17}43+x−8+8=17+8{\displaystyle 4^{3+x}-8+8=17+8}43+x=25{\displaystyle 4^{3+x}=25}
For example, if you take the base-10 log of both sides of 43+x=25{\displaystyle 4^{3+x}=25}, you would rewrite the equation like this: log43+x=log25{\displaystyle {\text{log}}4^{3+x}={\text{log}}25}.
For example, log43+x=log25{\displaystyle {\text{log}}4^{3+x}={\text{log}}25} can be rewritten as (3+x)log4=log25{\displaystyle (3+x){\text{log}}4={\text{log}}25}
For example, to isolate the x{\displaystyle x} in (3+x)log4=log25{\displaystyle (3+x){\text{log}}4={\text{log}}25}, you first need to divide each side of the equation by log4{\displaystyle {\text{log}}4}, then subtract 3 from both sides:(3+x)log4=log25{\displaystyle (3+x){\text{log}}4={\text{log}}25}(3+x)log4log4=log25log4{\displaystyle (3+x){\frac {{\text{log}}4}{{\text{log}}4}}={\frac {{\text{log}}25}{{\text{log}}4}}}3+x=log25log4{\displaystyle 3+x={\frac {{\text{log}}25}{{\text{log}}4}}}3+x−3=log25log4−3{\displaystyle 3+x-3={\frac {{\text{log}}25}{{\text{log}}4}}-3}x=log25log4−3{\displaystyle x={\frac {{\text{log}}25}{{\text{log}}4}}-3}
For example, to find log25{\displaystyle {\text{log}}25}, hit 25{\displaystyle 25}, then LOG{\displaystyle {\text{LOG}}} on your calculator, to get about 1. 3979. To find log4{\displaystyle {\text{log}}4}, hit 4{\displaystyle 4}, then LOG{\displaystyle {\text{LOG}}} on your calculator, to get about 0. 602. You new equation will now be x=1. 39790. 602−3{\displaystyle x={\frac {1. 3979}{0. 602}}-3}.
For example, in x=1. 39790. 602−3{\displaystyle x={\frac {1. 3979}{0. 602}}-3} you should divide first, then subtract:x=1. 39790. 602−3{\displaystyle x={\frac {1. 3979}{0. 602}}-3}x=2. 322−3{\displaystyle x=2. 322-3}x=−0. 678{\displaystyle x=-0. 678}.
