3 An exponential equation is an equation containing one or more expressions that have a variable as an exponent. To solve exponential equations:Try writing them so that the bases are the sameTake the logarithm of both sides
4 Rewrite the equation in the form 𝑏 𝑥 = 𝑏 𝑦 . Set 𝑥=𝑦. Solving Exponential Equations by Expressing Each Side as a Power of the Same BaseExpress each side as a power of the same base.Set the exponents equal to each other.Rewrite the equation in the form 𝑏 𝑥 = 𝑏 𝑦 .Set 𝑥=𝑦.Solve for the variable.
5 Solving Exponential Equations Solve and check.98 – x = 27x – 3Rewrite each side with the same base; 9 and 27 are powers of 3.(32)8 – x = (33)x – 3To raise a power to a power, multiply exponents.316 – 2x = 33x – 9Bases are the same, so the exponents must be equal.16 – 2x = 3x – 9x = 5Solve for x.
6 Check98 – x = 27x – 398 – – 3The solution is x = 5.
7 Solve and check: 𝟐 𝟑𝒙−𝟖 =𝟏𝟔 2 3𝑥−8 = 2 4 3𝑥−8=4 𝑥=4 Rewrite each side with the same base.2 3𝑥−8 = 2 4Now that the bases are the same, solve for 𝑥3𝑥−8=4𝑥=4
8 Check: 𝟐 𝟑𝒙−𝟖 =𝟏𝟔Substitute 4 for the variable.2 3(4)−8 =162 12−8 =162 4 =1616=16
9 Solve and check: 𝟓 𝟑𝒙−𝟔 =𝟏𝟐𝟓 5 3𝑥−6 = 5 3 3𝑥−6=3 𝑥=3 Rewrite each side with the same base.5 3𝑥−6 = 5 3Now that the bases are the same, solve for 𝑥3𝑥−6=3𝑥=3
10 Solve and check: 𝟓 𝟑𝒙−𝟔 =𝟏𝟐𝟓 5 3(3)−6 =125 5 9−6 =125 5 3 =125 125=125 Substitute 3 for the variable and solve.5 3(3)−6 =1255 9−6 =1255 3 =125125=125
11 Solving Exponential Equations Solve and check.4x – 1 = 5log 4x – 1 = log 5We cannot get common bases, so take the log of both sides.(x – 1)log 4 = log 5Apply the Power Property of Logarithms.x –1 =log5log4Divide both sides by log 4.x = ≈ 2.161log5log4Check Use a calculator.The solution is x ≈
12 Using Logarithms to Solve Exponential Equations Isolate the exponential expression.Take the natural logarithm on both sides of the equation for bases other than 10. Take the common logarithm on both sides of the equation for base 10.Simplify using one of the following properties:𝐥𝐧 𝒃 𝒙 =𝒙 𝐥𝐧 𝒃 or 𝐥𝐧 𝒆 𝒙 =𝒙 or 𝐥𝐨𝐠 𝟏𝟎 𝒙 =𝒙Solve for the equation.
13 Solve: 𝟒 𝒙 =𝟏𝟓ln 4 𝑥 = ln 15Take the natural logarithm on both sides𝑥 ln 4= ln 15Use the power rule𝑥 = ln 15 ln 4Solve for x by dividing both sides by 𝐥𝐧 𝟒𝒙≈𝟏.𝟗𝟓𝟑𝟒Use calculator.≈14.999≈15Check.
14 Solve: 𝑒 𝑥 =72 ln 𝑒 𝑥 = ln 72 𝑥 ln 𝑒= ln 72 𝑥= ln 72 𝑥≈4.277 Take the natural logarithm of both sides.ln 𝑒 𝑥 = ln 72𝑥 ln 𝑒= ln 72When you take the natural logarithm of base e, the ln e drops from the equation leaving only the exponent as seen above. This is using the inverse property 𝒍𝒏 𝒆 𝒙 =𝒙 . Also, ln 𝑒=1𝑥= ln 72𝑥≈4.277Check your answer.
15 Solve: 𝟒𝟎 𝒆 𝟎.𝟔𝒙 −𝟑=𝟐𝟑𝟕40 𝑒 0.6𝑥 =240Add 3 to both sides𝑒 0.6𝑥 =6Divide both sides by 40ln 𝑒 0.6𝑥 = ln 6Take the natural logarithm of both sides0.6𝑥= ln 6Use the inverse property 𝒍𝒏 𝒆 𝒙 =𝒙 .𝒙= 𝐥𝐧 𝟔 𝟎.𝟔 ≈ 2.99Divide both sides by 0.6 and solve for x
16 Solve 𝟓 𝒙−𝟐 = 𝟒 𝟐𝒙+𝟑ln 5 𝑥−2 = ln 4 2𝑥+3Take the natural logarithm on both sides𝑥−2 ln 5 =(2𝑥+3) ln 4Use the power rule𝑥 ln 5 −2 ln 5 =2𝑥 ln 4 +3 ln 4Use the distributive property𝑥 ln 5 −2𝑥 ln 4 =2 ln 5 +3 ln 4Rearrange terms𝑥( ln 5−2 ln 4)= 2 ln 5 +3 ln 4Factor out x2 ln 5+3 ln 4𝑥=ln 5−2 ln 4The solution is approximately −𝟔.𝟑𝟒
17 Solve: 𝒆 𝟐𝒙 −𝟒 𝒆 𝒙 +𝟑=𝟎 Let 𝒖= 𝒆 𝒙 𝑢 2 −4𝑢+3=0 Substitute 𝒖 for 𝒆 𝒙 𝑢−3 𝑢−1 =0Factor on the left𝑢−3=0 𝑜𝑟 𝑢−1=0Set each factor equal to 0𝑢= 𝑢=1Solve for 𝒖𝑒 𝑥 = 𝑒 𝑥 =1Replace 𝒆 𝒙 for 𝒖Take the natural logarithmof both sidesln 𝑒 𝑥 = ln ln 𝑒 𝑥 = ln 1𝒙= 𝐥𝐧 𝟑 ≈𝟏.𝟏𝟎 𝒙=𝟎
18 Using the Definition of a Logarithm to Solve Logarithmic Equations Express the equation in the form log 𝑏 𝑀=𝑐.Use the definition of a logarithm to rewrite the equation in exponential form.𝐥𝐨𝐠 𝐛 𝐌=𝐜 means 𝐛 𝐜 =𝐌Solve for the variable.Check your solutions for 𝑀>0 in the original equation.
19 Solve and check: 𝐥𝐨𝐠 𝟒 𝒙+𝟑 =𝟐 4 2 =𝑥+3Rewrite in exponential form16=𝑥+3𝟏𝟑=𝒙
20 Solve and check: 𝟑 𝐥𝐧 𝟐𝒙 =𝟏𝟐ln 2𝑥 =4Divide both sides by 3Rewrite the natural logarithm showing base elog 𝑒 2𝑥 =4𝑒 4 =2𝑥Rewrite in exponential form𝒆 𝟒 𝟐 =𝒙≈𝟐𝟕.𝟑Divide both sides by 2
21 Solve and check: 𝐥𝐨𝐠 𝟐 𝒙 + 𝐥𝐨𝐠 𝟐 𝒙−𝟕 =𝟑 log 2 [𝑥 𝑥−7 ]=3Use the product rule2 3 =𝑥(𝑥−7)Rewrite in exponential form8= 𝑥 2 −7𝑥Simplify𝑥 2 −7𝑥−8=0Set up as a quadratic equation𝑥−8 𝑥+1 =0Factor𝑥−8=0 𝒙=𝟖 𝑥+1= 𝒙=−𝟏Always check your answers with original equation.
22 If 𝐥𝐨𝐠 𝒃 𝑴= 𝐥𝐨𝐠 𝒃 𝑵 , then 𝑴=𝑵. Using the One-to-One Property of Logarithms to Solve Logarithmic EquationsExpress the equation in the form log 𝑏 𝑀= log 𝑏 𝑁. The coefficient must be equal to 1 on both sides.Use the one-to-one property to rewrite the equation without the logarithm.If 𝐥𝐨𝐠 𝒃 𝑴= 𝐥𝐨𝐠 𝒃 𝑵 , then 𝑴=𝑵.Solve for the variable.Check proposed solutions in the original equation. 𝑀 and 𝑁 must be positive.
23 ( 𝑥+2 4𝑥+3 ) = ln ( 1 𝑥 ) ln Solve: 𝐥𝐧 𝒙+𝟐 − 𝐥𝐧 𝟒𝒙+𝟑 = 𝐥𝐧 ( 𝟏 𝒙 ) Use the quotient rule𝑥+2 4𝑥+3 = 1 𝑥Use the one-to-one property𝑥 𝑥+2 =1(4𝑥+3)Cross multiply𝑥 2 +2𝑥=4𝑥+3Use distributive property𝑥 2 +2𝑥−4𝑥−3=0Set up as a quadratic equation𝑥 2 −2𝑥−3=0
24 Check by substituting each solution into the original equation. 𝑥 2 −2𝑥−3=0(𝑥−3)(𝑥+1)Factor𝑥−3= 𝑥+1=0Set each factor equal to 0𝒙=𝟑 𝒙=−𝟏Check by substituting each solution into the original equation.𝐥𝐧 𝒙+𝟐 − 𝐥𝐧 𝟒𝒙+𝟑 = 𝐥𝐧 ( 𝟏 𝒙 )