Question

Express the following using indices: (a) \(\log 35=1.5441\) (b) \(\log _{6} 1296=4\) (c) \(\ln 50=3.9120\) (d) \(\log _{9} 3=0.5\)

Short answer

Question: Express the given logarithmic expressions using indices. (a) \(\log 35=1.5441\) (b) \(\log_{6} 1296=4\) (c) \(\ln 50=3.9120\) (d) \(\log_{9} 3=0.5\) Answer: (a) \(10^{1.5441} = 35\) (b) \(6^4 = 1296\) (c) \(e^{3.9120} = 50\) (d) \(9^{0.5} = 3\)

Step-by-step solution

(a) Transform logarithm to exponential form

Given the expression, \(\log 35=1.5441\). This is a base 10 logarithm, so we can rewrite it as \(10^{1.5441} = 35\).

(b) Transform logarithm to exponential form

Given the expression, \(\log_{6} 1296=4\). Using the logarithm property mentioned earlier, we have \(6^4 = 1296\).

(c) Transform logarithm to exponential form

Given the expression, \(\ln 50=3.9120\). Here, \(\ln\) denotes the natural logarithm, which is a base \(e\) logarithm. Therefore, we can write this as \(e^{3.9120} = 50\).

(d) Transform logarithm to exponential form

Given the expression, \(\log_{9} 3=0.5\). Using the logarithm property, we get \(9^{0.5} = 3\). So our expressions in indices are as follows: (a) \(10^{1.5441} = 35\) (b) \(6^4 = 1296\) (c) \(e^{3.9120} = 50\) (d) \(9^{0.5} = 3\)

Key concepts

Logarithmic Equations

Logarithmic equations are mathematical expressions that describe the relationship between a number and the power to which a given base must be raised to obtain that number. In simpler terms, a logarithm answers the question: 'To what exponent must we raise a base to get a certain number?'
Take the logarithmic equation \(\log 35=1.5441\) as an example. Converting it to exponential form helps us understand it better. This conversion is straightforward: the base of the logarithm, which is 10 if not specified (known as a common logarithm), is raised to the power of the result to equal the number. In this case, \(10^{1.5441} = 35\). This illustrates that 10 raised to the power of 1.5441 gives us the number 35.
Similarly, for \(\log_{6} 1296=4\), we convert the logarithm to the exponential form by taking the base 6 and raising it to the power 4, which equals 1296 as expressed by \(6^4 = 1296\). This implies that raising 6 to the fourth power yields 1296.
In solving logarithmic equations, it is crucial to remember the correspondence between logarithmic and exponential forms. Understanding this relationship helps students transition seamlessly between logarithms and exponents, crucial for tackling a range of problems in mathematics.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a special type of logarithm whose base is the mathematical constant \(e\), approximately equal to 2.71828. It is a fundamental concept in various branches of mathematics, including calculus, due to its unique properties where the rate of growth is proportional to the quantity itself.
When dealing with natural logarithms, such as in the expression \(\ln 50=3.9120\), we are essentially asking the question: 'To what power must \(e\) be raised to obtain 50?' The exponential form of this natural logarithm is \(e^{3.9120} = 50\), which exhibits the power to which \(e\) must be raised to result in 50.
What sets the natural logarithm apart is its intimate connection with exponential growth and decay processes. This makes \(\ln\) extremely valuable in describing natural phenomena, from population modeling to radioactive decay, and in financial contexts, where compound interest is calculated.

Exponential Expressions

Exponential expressions, such as those arising from converting logarithmic forms, represent numbers as the result of raising a base to a certain exponent or power. The general form of an exponential expression is \(b^x\), where \(b\) is the base and \(x\) is the exponent.
Consider the exponential expression \(9^{0.5} = 3\) derived from the logarithmic equation \(\log_{9} 3=0.5\). It indicates that the square root of 9 is 3, as the exponent 0.5 corresponds to the square root function.
Understanding exponential expressions is key to mastering various scientific and mathematical concepts, from population growth and radioactive decay to the function of interest rates in finance. When students become comfortable with transforming logarithms to their exponential counterparts, it can also facilitate a deeper comprehension of equations in physics, particularly those involving exponential decay or growth.