The term ‘log 5’ refers to the logarithm of the number 5. In mathematics, a logarithm is essentially the opposite of exponentiation. It answers the question: to what exponent must a base (often 10 or e) be raised to produce a given number? Specifically, ‘log 5’ usually implies the base 10 logarithm (log10 5), which means the power to which 10 must be raised to yield 5.
To find logs of numbers without a calculator, you can use several techniques:
- Logarithm Tables: Before calculators became common, mathematicians often referred to logarithm tables that listed the logarithms of many numbers. You can look up ‘5’ in these tables to find its logarithm.
- Known Values: Remember some key logarithm values. For example, log10 1 = 0, log10 10 = 1, log10 100 = 2. Knowing these can help you estimate logs for numbers close to these known values.
- Change of Base Formula: If you know the logs of other numbers, you can use the change of base formula: logb(a) = logc(a) / logc(b), where ‘c’ is any other base. This allows you to manipulate numbers you’re more comfortable with.
- Estimation: You can also approximate logarithms using interpolation. For example, log10 4 is about 0.6 (since it’s close to log10 1 and log10 10), and log10 6 is a bit more than 0.78. Knowing this, you can estimate log10 5 to be around 0.7.
In conclusion, while using a calculator is the most straightforward method to find logarithms, you can utilize tables, known values, and basic properties of logarithms to estimate them without electronic help.