For variables and , plotting against gives a straight-line graph passing through the points and .
Show that = where and are integers to be found.
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For variables and , plotting against gives a straight-line graph passing through the points and .
Show that = where and are integers to be found.
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Variables and are such that, when is plotted against , a straight line graph passing through the points (6,7) and (10,9) is obtained. Find as a function of .
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Given that , find the value of .
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Variables and are connected by the relationship , where and are constants.
Transform the relationship to straight line form.
When is plotted against a straight line graph passing through the points (0, 0.5) and (3.2, 1.7) is obtained.
Find the value of and of .
Find the value of when .
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The population , in millions, of a country is given by , where is the number of years after January 2000 and and are constants. In January 2010 the population was 40 million and had increased to 45 million by January 2013.
Show that 1.04 to 2 decimal places and find to the nearest integer.
Find the population in January 2020, giving your answer to the nearest million.
In January of which year will the population be over 100 million for the first time?
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The number, , of bacteria in a sample is given by , where and are constants and is time in weeks. Initially there are 500 bacteria which increase to 600 after 1 week.
Find the value of and of .
Find the number of bacteria present after 2 weeks.
Find the first week in which the number of bacteria is greater than 1 000 000.
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Find the exact solution of .
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Given that , find the value of.
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In this question, and are positive constants.
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The function is defined by.
Sketch the graph of and hence sketch the graph of on the axes below.
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Solve the equation
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Write as a single logarithm.
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Using the substitution , or otherwise, solve .
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Solve the equation .
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Solve the equation .
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Write the expression in the form , where and are integers.
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Find the exact solution of .
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Solve the simultaneous equations
,
,
giving and in exact simplified form.
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.
Solve this equation for , giving your answers in terms of .
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Solve the simultaneous equations.
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It is known that , where and are constants. When is plotted against , a straight line passing through the points (3.63, 5.25) and (4.83, 6.88) is obtained.
Find the value of and of .
Using your values of and , find the value of when ,
Find the positive value of when .
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Show that .
Find the roots of .
Give a reason why only one root is a valid solution of the logarithmic equations. Find the value of corresponding to this root.
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