Question
(i) Show that the equation \(log_{10}(x-4)=2-log_{10}x\) can be written as a quadratic equation in x.
(ii) Hence solve the equation \(log_{10}(x-4)=2-log_{10} \) x giving your answer correct to 3 significant figures.
Answer/Explanation
1(i) Use law for the logarithm of a product or quotient
\(Use log_{10}100=2 or 10^{2}=100\)
obtain\( x^{2}-4x-100=0\) , or equivalent
1(ii) Solve a 3-term quadratic equation
Obtain answer 12.2 only
Question
Showing all necessary working, solve the equation ln(2x − 3 )= 2 ln x − ln (x − 1). Give your answer correct to 2 decimal places.
Answer/Explanation
Use law for the logarithm of a product ,quotient or power
Obtain a correct equation free of logarithms
Solve a 3-term quadratic obtaining at least one root
Obtain answer x=4.30 only
Question
It is given that 2 ln(4x − 5 )+ ln(x + 1 )3 ln 3.(i) Show that \(16x^{3}-24x^{2}-15x-2=0\)
(ii) By first using the factor theorem, factorise \(16x^{3}-24x^{2}-15x-2=0\) completely.
(iii) Hence solve the equation 2 ln (4x − 5 )+ ln(x + 1) = 3 ln 3.
Answer/Explanation
(i) Use law for the logarithm for a product or quotient or exponentiation AND for a power
Obtain\( (4x-5)^{2}(x+1)=27\)
Obtain given equation correctly \(16x^{3}-24x^{2}-15x-2=0\)
(ii) Obtain x = 2 is root or (x – 2) is a factor, or likewise with\( x=-\frac{1}{4}\)
Divide by (x – 2) to reach a quotient of the form\((16x^{3}+kx)\)
Obtain quotient \(16x^{3}+8x+1\)
(iii) State x = 2 only
Question
(i) Show that the equation
\(log_{10}\) (x-4)=2-\(log_{10}x\)can be written as a quadratic equation in x.
(ii) Hence solve the equation giving your answer correct to 3 significant figures.
Answer/Explanation
(i) Use law for the logarithm of a product or quotient Use \(log_{10}100=2or 10^{2}=100\)Obtain \(x^{2}-4x-100=0\),or equivalent
(ii) Solve a 3-term quadratic equation Obtain answer 12.2 only