Page 1 Examples of Solving Exponential Equations Example ...

Example – Solve: 42x 3 1+ æ ö= ç ÷è 32 ø3x+142x 3 1+ æ ö= ç ÷è 32 ø3x+1Determine if 4 and 1/32 can be written using the same base. In thiscase both 4 and 1/32 can be written using the base 2.2 - 5( 2 ) = ( 2 )2x+ 3 3x+1Rewrite the problem using the same base. Note that 1/32 = 1/2 5 =2 –5 .4x+ 6 -15x-52 = 24x + 6 = -15x - 511x = -19Use the properties of exponents to simplify the exponents, when apower is raised to a power, we multiply the powers.Since the bases are the same, we can drop the bases and set theexponents equal to each other.Finish solving by subtracting 6 from each side, adding 15x to eachside, and then dividing each side by 19.Therefore, the solution to the problem 42x 3 1+ æ ö= ç ÷è 32 ø3x+111is x = - .19Example – Solve:4-7xe = 8714-7xe = 8714-7x( )ln e= ln(871)(4 - 7x)(ln e) = ln 871(4 - 7x)(1) = ln 8714 - 7x » 6.769642x » -0.395663Determine if e and 871 can be written using the same base. In thiscase e and 871 cannot be written using the same base, so we mustuse logarithms.Take the common logarithm or natural logarithm of each side. Inthis case, we should use the natural logarithm because the base is e.Use the properties of logarithms to rewrite the problem. Move theexponent out front which turns this into a multiplication problem.Use the properties of logarithms to find ln e. Property 2 states thatln e = 1.Use a calculator to find ln 871. Round the answer as appropriate,these answers will use 6 decimal places.Finish solving the problem by subtracting 4 from each side andthen dividing each side by –7.Therefore, the solution to the problem 4 -e 7x = 871 is x ≈ –0.395663.

4x+3Example – Solve: 7 = 5234x+37 = 5234x + 3log(7 ) = log(523)Determine if 7 and 523 can be written using the same base. In thiscase 7 and 523 cannot be written using the same base, so we mustuse logarithms.Take the common logarithm or natural logarithm of each side.(4x + 3)(log 7) = log 523log5234x + 3 =log 74x + 3 » 3.216789x » 0.054197Use the properties of logarithms to rewrite the problem. Move theexponent out front which turns this into a multiplication problem.Divide each side by log 7.Use a calculator to find log 523 divided by log 7. Round theanswer as appropriate, these answers will use 6 decimal places.Finish solving the problem by subtracting 3 from each side andthen dividing each side by 4.Therefore, the solution to the problem 4x +7 3 = 523 is x ≈ 0.054197.Example – Solve:5x-93e = 292e5x-9 292ln e=( )35x 9 292- = lnæ ç ö÷è 3 øTo solve this problem we need to divide by 3 first, and thendetermine if e and 292/3 can be written using the same base. In thiscase e and 292/3 cannot be written using the same base, so wemust use logarithms.Take the common logarithm or natural logarithm of each side. Inthis case, we should use the natural logarithm because the base is e.æ 292 ö(5x - 9)(ln e) = ln ç ÷è 3 øUse the properties of logarithms to rewrite the problem. Move theexponent out front which turns this into a multiplication problem.æ 292 ö(5x - 9)(1) = ln ç ÷è 3 øUse the properties of logarithms to find ln e. Property 2 states thatln e = 1.5x - 9 » 4.578142x » 2.715628Use a calculator to find ln(292/3). Round the answer asappropriate, these answers will use 6 decimal places.Finish solving the problem by adding 9 to each side and thendividing each side by 5.5x-9Therefore, the solution to the problem 3e = 292 is x ≈ 2.715628.