Logarithmic Equations Worksheet

Pre-Calculus/Trig 3 Name: KEY UNIT 3: Exponential & Logarithmic Functions - SECTION 7 WORKSHEET Date: SOLVING LOGARITHMIC EQUATIONS Directions: Solve each logarithmic equation. Remember to check for extraneous solutions! 1.) (log(5x)=log(2x+9)) 1 2.) (log(10-4x)=log(10-3x))   1. www.docsity.com www.docsity.com \(5x=2x+9\) \(10-4x=10-3x\) \(3x=9\) \(-x=0\) \(x=3\) ✓ \(x=0\) ✓ 3.) (log(4p-2)=log(-5p+5)) 4.) (log(4k-5)=log(2k-1)) \(4p-2=-5p+5\) \(4k-5=2k-1\) \(9p=7\) \(2k=4\) \(p=\frac{7}{9}\) ✓ \(k=2\) ✓ 5.) (log(-2a+9)=log(7-4a)) 6.) (2~log_{7}(-2r)=0) \(-2a+9=7-4a\) \(log_{7}(-2r)=0\) \(2a=-2\) \(7^0=-2r\) \(a=-1\) ✓ \(1=-2r\) \(r=-\frac{1}{2}\) ✓ 7.) (-10+log_{3}(n+3)=-10) 8.) (-2~log_{5}(7x)=2) \(log_{3}(n+3)=0\) \(log_{5}(7x)=-1\) \(3^0=n+3\) \(5^{-1}=7x\) \(1=n+3\) \(\frac{1}{5}=7x\) \(n=-2\) ✓ \(x=\frac{1}{35}\) ✓ 9.) (log(-m)+2=4) 10.) (-6~log_{3}(x-3)=-24) \(log(-m)=2\) \(log_{3}(x-3)=4\) \(10^2=-m\) \(3^4=x-3\) \(100=-m\) \(81=x-3\) \(m=-100\) ✓ \(x=84\) ✓ 11.) (log_{12}(v^{2}+35)=log_{12}(-12v-1)) 12.) (log_{9}(-11x+2)=log_{9}(x^{2}+30)) 13.) (log(16+2b)=log(b^2-4b)) 14.) (ln(x^2+12)=ln(-9x-2)) \(16+2b=b^2-4b\) \(x^2+12=-9x-2\) \(0=b^2-6b-16\) \(x^2+9x+14=0\) \(0=(b-8)(b+2)\) \((x+7)(x+2)=0\) \(b=8\) ✓ \(b=-2\) ✓ \(x=-7\) ✓ \(x=-2\) ✓ 15.) (log~x+log~8=2) 16.) (log~x-log~2=1) \(log(8x)=2\) \(log(\frac{x}{2})=1\) \(10^2=8x\) \(10^1=\frac{x}{2}\) \(100=8x\) \(10=\frac{x}{2}\) \(x=\frac{100}{8}\) \(x=20\) ✓ \(x=\frac{25}{2}\) ✓ 17.) (log~2+log~x=1) 18.) (log~x+log~7=log~37) \(log(2x)=1\) \(log(7x)=log~37\) \(10^1=2x\) \(7x=37\) \(10=2x\) \(x=\frac{37}{7}\) ✓ \(x=5\) ✓ 19.) (log_{8}2+log_{8}(4x^{2})=1) 20.) (log_{9}(x+6)-log_{9}x=log_{9}2) \(log_{8}(8x^{2})=1\) \(log_{9}(\frac{x+6}{x})=log_{9}2\) \(8^1=8x^2\) \(\frac{x+6}{x}=2\) \(8=8x^2\) \(x+6=2x\) \(x^2=1\) \(x=6\) ✓ \(x=\pm 1\) ✓ 21.) (log_{6}(x+1)-log_{6}x=log_{6}29) 22.) (log_{5}6+log_{5}(2x^{2})=log_{5}48) \(log_{6}(\frac{x+1}{x})=log_{6}29\) \(log_{5}(12x^{2})=log_{5}48\) \(\frac{x+1}{x}=29\) \(12x^2=48\) \(x+1=29x\) \(x^2=4\) \(1=28x\) \(x=\pm 2\) ✓ \(x=\frac{1}{28}\) ✓ 23.) (ln(4x-1)=3) 24.) (ln(3x+11)=4) \(e^3=4x-1\) \(e^4=3x+11\) \(e^3+1=4x\) \(e^4-11=3x\) \(\frac{e^3+1}{4}=x\) \(\frac{e^4-11}{3}=x\) \(x \approx 5.2714\) ✓