11.08 Equations using logarithms (Enrichment)
Solve for x in each of the following logarithmic equations:
\log_{4} 9 x = 2
9 \log x = 45
\log_{64} x = \dfrac{1}{3}
\log_{2} \left( 5 x - 1\right) = 6
\log_{7} \left( 10 x + 2311\right) = 4
\log_{9} \left( 5 x - 32\right) = \log_{9} 8
\log_{7} \left( 6 x - 6\right) = \log_{7} \left( 3 x - 2\right)
\log_{3} \left( 5 x - 2\right) = \log_{3} \left( 4 x + 2\right)
\log_{10} x + \log_{10} 9 = \log_{10} 45
\log \left(x + 3\right) = \log x + \log 6
\log_{10} 6 + \log_{10} x = \log_{10} \left(x + 8\right)
\log \left( 11 x - 2\right) - \log \left( 4 x - 2\right) = \log 3
\log \left(x + 30\right) - \log \left(x + 2\right) = \log x
\log_{9} \left( 4 x + 12\right) = \log_{9} \left(x + 7\right) - \log_{9} 7
Solve \log_{8} y = 4 for y.
Solve for the exact value of x in each of the following exponential equations:
3 \left(10^{x}\right) = 6
\dfrac{1}{7} \left(2^{x}\right) = 3
3 \left(5^{ 4 x + 5}\right) + 2 = 5
2 \times 3^{ 5 x} + 4 = 16
Solve for the exact value of y in each of the following exponential equations:
2^{ 5 y + 2} = 3^{4 - 3 y}
3^{y + 5} = 6^{ 2 y}
Solve for x in each of the following exponential equations, correct to two decimal places:
5^{x} = 9
\left(\dfrac{1}{4}\right)^{x + 3} = \sqrt{7}
3^{x} = 2^{x + 1}
5^{x + 4} = 25^{ 8 x - 4}
Find the solution to 3^{x} = - 2.
Solve for y in each of the following exponential equations, correct to two decimal places:
12^{y} = 23
3^{y + 1} = 6
4^{y - 3} = 25
4^{y + 1} = 7
7^{ 4 y + 3} = 6
6^{y + \left( - 3 \right)} = 11^{y}
6^{y + 5} = 5^{y + 2}
Solve the following exponential equations:
9^{y} = 27
\left(\sqrt{6}\right)^{y} = 36
\left(\sqrt{2}\right)^{k} = 0.5
3^{ 5 x - 10} = 1
\dfrac{1}{3^{x - 3}} = \sqrt[3]{9}
\left(\dfrac{1}{9}\right)^{x + 5} = 81
\left(\dfrac{1}{8}\right)^{x - 3} = 16^{ 4 x - 3}
\dfrac{25^{y}}{5^{4 - y}} = \sqrt{125}
8^{x + 5} = \dfrac{1}{32 \sqrt{2}}
30 \times 2^{x - 6} = 15
2^{x} \times 2^{x + 3} = 32
81^{x - 1} = 9^{ 3 x + 5}
25^{x + 1} = 125^{ 3 x - 4}
a^{x + 1} = a^{3} \sqrt{a}
3^{x^{2} - 3 x} = 81
27 \left(2^{x}\right) = 6^{x}
Sketch the graph of the inverse of the following functions:
Find the x-coordinate of the point of intersection of the graphs of y = 2^{ 5 x} and y = 4^{x - 3}.
Find the value of h, given the point \left(h,\dfrac{1}{9}\right) lies on the curve y = 3^{ - x }.
Given the points \left(3, n\right), \left(k, 16\right) and \left(m, \dfrac{1}{4}\right) all lie on the curve with equation y = 2^{x}, find the value of:
n
k
m
A certain type of cell splits in two every hour and each cell produced also splits in two each hour. The total number of cells after t hours is given by:
N(t)=2^tFind the time when the number of cells will reach the following amounts:
1024
4096
The frequency f \left(\text{Hz}\right) of the nth key of an 88-key piano is given by f \left( n \right) = 440 \left(2^{\frac{1}{12}}\right)^{n - 49}.
Find the frequency of the forty-ninth key.
Find the frequency of the 40th key to the nearest whole number.
Find the value of n that corresponds to the key with a frequency of 1760 \text{ Hz}.