Solve the following equations.

(a) \(\log_5(5x-2)-\log_{25}x=\frac12\)

(b) \(\mathrm{e}^{3y-7}+\frac{4}{\mathrm{e}^3}=\frac{5}{\mathrm{e}^{3y-1}}\)

(a) Write \(5\lg a-4\lg b-3\) as a single base 10 logarithm.

(b) Solve the equation \(\log_5(x+1)-\log_{x+1}5=0\).

(a) Solve the equation \(\log_2x-4=5\log_x2\).

(b) Solve the equation \(\mathrm{e}^{x^2-3}=25\mathrm{e}^{7-x^2}\), giving your answers in exact form.

Solve the following equations. (a) \(\log _{2} x^{2}+\log _{16} x=18\) (b) \(\mathrm{e}^{2 x+1}-10 \mathrm{e}^{-2 x-1}=3\)

(a) Given that \(\log _{p} a+\log _{p} 12-\log _{p} 6=3 \log _{p} 4\), find the value of \(a\).

(b) Find the exact solutions of the equation \(4 \log _{3} x=9 \log _{x} 3\).

(a) Write \(1+\lg \left(x^{2}-1\right)-2 \lg (x-1)\), where \(x\gt 1\), as a single logarithm to base 10 . Give your answer in its simplest form. (b) Solve the equation \(4 \log _{5}(x+1)=9 \log _{(x+1)} 5\), giving your answers in the form \(a+b \sqrt{c}\), where \(a, b\) and \(c\) are constants.

Solve the following simultaneous equations, giving your answers in exact form. \(\begin{array}{c} 8 \log _{3} x+12 \log _{81} y=5 \\ 4 \log _{9} x+3 \log _{3} y=2 \end{array}\)

(a) Solve the following simultaneous equations.

\(3\log_2x+2\log_2y=24\)

\(5\log_2x-3\log_2y=2\)

(b) Solve the equation

\(\frac{2^{t+4}}{2^{1-2t}}=512\).

Solve the following simultaneous equations.

\(5x-3\ln y=2\)

\(x+\ln y=1\)

Solve the following equations.

(a) \(\displaystyle \frac{(\mathrm e^{x+1})^2}{\sqrt{\mathrm e^x}}=10\)

(b) \(2\log_9y-\log_9(4y-9)=\frac12\)

(a) Write

\(3\lg x-2\lg y^2-3\)

as a single logarithm to base 10.

(b) Solve the equation

\(\log_3 x+\log_x 3=\frac52.\)

(a) Find the exact solutions of the equation

\(6p^{\frac13}-5p^{-\frac13}-13=0.\)

(b) Solve the equation

\(2\lg(2x+5)-\lg(x+2)=1,\)

giving your answers in exact form.

(a) Write

\(3+2\lg a-\frac12\lg(4b^2),\)

where \(a\) and \(b\) are both positive, as a single logarithm to base 10. Give your answer in its simplest form.

(b) Given that \(2\log_c3=7+4\log_3c\), find the possible values of the positive constant \(c\), giving your answers in exact form.

(a) Write \(3\lg x-\frac12\lg4+2\) as a single logarithm to base 10.

(b) Solve the equation

\(2\log_a4-3\log_4a-5=0\),

giving your answers exactly.

Solve the equation

\(\log_5(8x+7)-\log_5(2x)=2.\)

A function \(f\) is such that \(f(x)=\ln(2x+1)\), for \(x\gt -\frac12\).

(a) Write down the range of \(f\).

A function \(g\) is such that \(g(x)=5x-7\), for \(x\in\mathbb R\).

(b) Find the exact solution of \(gf(x)=13\).

(c) Find the solution of \(f'(x)=g^{-1}(x)\).

(a) Given the simultaneous equations

\(\lg x+2\lg y=1,\)

\(x-3y^2=13,\)

(i) show that \(x^2-13x-30=0\).

(ii) Solve these simultaneous equations, giving your answers in exact form.

(b) Solve the equation

\(\log_a x+3\log_x a=4,\)

where \(a\) is a positive constant, giving \(x\) in terms of \(a\).

(a) Write \(2\lg x-\{\lg(x+6)+\lg 3\}\) as a single logarithm to base 10.

(b) Hence solve the equation \(2\lg x-\{\lg(x+6)+\lg 3\}=0\).

(a) Solve the following simultaneous equations.

\(3y-2x+2=0,\qquad xy=\frac12.\)

(b) Solve the equation

\(\log_3x+3=10\log_x3,\)

giving your answers as powers of \(3\).

Solve the equation

\(\log_3(11x-8)=1+\frac{2}{\log_x3},\)

given that \(x\gt 1\).

Solve the equation

\(\lg(2x-1)+\lg(x+2)=2-\lg 4.\)

Find the exact solutions of the equation

\(3(\ln 5x)^2+2\ln 5x-1=0.\)

(a) Given that

\(\log_a p+\log_a 5-\log_a 4=\log_a 20,\)

find the value of \(p\).

(b) Solve the equation

\(3^{2x+1}+8(3^x)-3=0.\)

(c) Solve the equation

\(4\log_y 2+\log_2 y=4.\)

In this question, \(a\), \(b\), \(c\) and \(d\) are positive constants.

(a)

(i) It is given that

\(y=\log_a(x+3)+\log_a(2x-1).\)

Explain why \(x\) must be greater than \(\frac12\).

(ii) Find the exact solution of the equation

\(\frac{\log_a6}{\log_a(y+3)}=2.\)

(b) Write the expression

\(\log_a9+(\log_a b)\bigl(\log_{\sqrt b}9a\bigr)\)

in the form \(c+d\log_a9\), where \(c\) and \(d\) are integers.

(a) Write

\(\frac{x(27xy^3)^{\frac53}}{\sqrt[4]{81y^5}}\)

in the form \(3^a x^b y^c\), where \(a\), \(b\) and \(c\) are constants.

(b)

(i) Find the value of \(a\) such that

\(2\log_a8=\frac32.\)

(ii) Write \(\log_{a^2}3a\) as a single logarithm to base \(a\).

(a) Using logarithms, solve the equation

\(5^{x-2}=3\times 2^{2x+3}.\)

Give your answer correct to 1 decimal place.

(b) Solve the equation

\(\log_3(y^2+11)-2=\log_3(y-1).\)

(a) Solve the equation

\(\log_6(2x-3)=\frac12.\)

Give your answer in exact form.

(b) Solve the equation

\(\ln 2u-\ln(u-4)=1.\)

Give your answer in exact form.

(c) Solve the equation

\(\frac{3^v}{27^{2v-5}}=9.\)

Solve the simultaneous equations

\(\log_3(x+y)=2,\)

\(2\log_3(x+1)=\log_3(y+2).\)

Do not use a calculator in this question.

It is given that

\(\log_2(y+1)=3-2\log_2 x\)

and

\(\log_2(x+2)=2+\log_2 y.\)

(a) Show that \(x^3+6x^2-32=0\).

(b) Find the roots of the equation \(x^3+6x^2-32=0\).

(c) Hence find the value of \(y\).

(a) Solve \(\log_3 x+\log_9 x=12\).

(b) Solve \(\log_4(3y^2-10)=2\log_4(y-1)+\frac12\).

(a) Write

\(\frac{\sqrt{p}\left(\frac{qp}{r}\right)^2}{p^{-1}\sqrt[3]{qr}}\)

in the form \(p^aq^br^c\), where \(a\), \(b\) and \(c\) are constants.

(b) Solve

\(\log_7x+2\log_x7=3.\)

(a) Solve

\(e^{2x+1}=3e^{4-3x}.\)

(b) Solve

\(\lg(y-6)+\lg(y+15)=2.\)

(a) Solve the following equations.

(i) \(5e^{3x+4}=14\).

(ii) \(\lg(2y-7)+\lg y=2\lg3\).

(b) Write \(\dfrac{\log_2p-\log_2q}{(\log_2r)(\log_r2)}\) as a single logarithm to base \(2\).

(a)(i) Solve \(\lg x=3\).

(a)(ii) Write \(\lg a-2\lg b+3\) as a single logarithm.

(b)(i) Solve \(x-5+\dfrac6x=0\).

(b)(ii) Hence, showing all your working, find the values of \(a\) such that

\(\log_4 a-5+6\log_a4=0.\)

Solve the simultaneous equations

\(\log_2(x+2y)=3, \qquad \log_2(3x)-\log_2y=1.\)

Solve the equation \(\log_5(10x+5)=2+\log_5(x-7)\).

Solve the simultaneous equations

\(\log_2(x+4)=2\log_2y,\)

\(\log_2(7y-x)=4.\)

Solve the simultaneous equations

\(\log_3(x+1)=1+\log_3 y,\)

\(\log_3(x-y)=2.\)