1. Condense $5 \log _3(x)+2 \log _3(4 x)-\log _3 8 x^5$
2. Prove the hyperbolic identity $ch ^2- sh ^2=1$
3. A cylindrical container has a radius of 7 m and a vertical height of 20 m.
a) Calculate the amount of water the container can hold in litres.
b) If the container has a mass of 1320 kg, calculate the density of the material used in making the container.
4. The following marks were scored by engineering students in mathematics test
| Marks | $40-44$ | $45-49$ | $50-54$ | $55-59$ | $60-64$ | $65-69$ | $70-74$ |
| :-------- | :------ | :------ | :------ | :------ | :------ | :------ | :------ |
| frequency | 3 | 8 | 12 | 15 | 19 | 8 | 9 |
State the;
I) Modal class
II) Modal frequency
Calculate the;
I) Mean mark
II) Median mark
III) Variance
5. Determine the modulus and argument of the complex number
$z=\frac{9+3 j}{1+2 j}$
Answer (1)
Use the power rule to simplify the coefficients of the logarithms: 5 lo g 3 ( x ) = lo g 3 ( x 5 ) and 2 lo g 3 ( 4 x ) = lo g 3 ( 16 x 2 ) .
Apply the product rule to combine the first two terms: lo g 3 ( x 5 ) + lo g 3 ( 16 x 2 ) = lo g 3 ( 16 x 7 ) .
Use the quotient rule to condense the expression: lo g 3 ( 16 x 7 ) − lo g 3 ( 8 x 5 ) = lo g 3 ( 8 x 5 16 x 7 ) .
Simplify the expression to obtain the final condensed form: lo g 3 ( 2 x 2 ) .
Explanation
Problem Analysis We are asked to condense the logarithmic expression 5 lo g 3 ( x ) + 2 lo g 3 ( 4 x ) − lo g 3 8 x 5 . We will use the properties of logarithms to simplify the expression.
Applying Power Rule Using the power rule of logarithms, a lo g b ( x ) = lo g b ( x a ) , we have:
5 lo g 3 ( x ) = lo g 3 ( x 5 ) and 2 lo g 3 ( 4 x ) = lo g 3 (( 4 x ) 2 ) = lo g 3 ( 16 x 2 ) .
So the expression becomes:
lo g 3 ( x 5 ) + lo g 3 ( 16 x 2 ) − lo g 3 ( 8 x 5 ) .
Applying Product Rule Using the product rule of logarithms, lo g b ( x ) + lo g b ( y ) = lo g b ( x y ) , we have:
lo g 3 ( x 5 ) + lo g 3 ( 16 x 2 ) = lo g 3 ( x 5 "."16 x 2 ) = lo g 3 ( 16 x 7 ) .
So the expression becomes:
lo g 3 ( 16 x 7 ) − lo g 3 ( 8 x 5 ) .
Applying Quotient Rule Using the quotient rule of logarithms, lo g b ( x ) − lo g b ( y ) = lo g b ( y x ) , we have:
lo g 3 ( 16 x 7 ) − lo g 3 ( 8 x 5 ) = lo g 3 ( 8 x 5 16 x 7 ) = lo g 3 ( 2 x 2 ) .
Therefore, the condensed expression is lo g 3 ( 2 x 2 ) .
Final Answer The condensed form of the given logarithmic expression is lo g 3 ( 2 x 2 ) .
Examples
Logarithmic scales are used to represent large ranges of values in a compact way. For example, the Richter scale for earthquake magnitudes uses logarithms to quantify the energy released by an earthquake. Similarly, in chemistry, pH values are logarithmic measures of the concentration of hydrogen ions in a solution. Understanding how to condense and manipulate logarithmic expressions is crucial in these fields for simplifying calculations and interpreting data effectively.
