Logarithmic Equations

An equation involving logarithms with unknown(s) is called a logarithmic equation. For example: and log (x - 1) + log 5 = 3 are logarithmic equations. log (2x) = 1, log (x + 4) = log 9 You can solve these logarithmic equations using the definition and properties of logarithms.

Check the answer obtained. Take log (2x) = 1 as an example. ∵ log (2x) = 1 If log y = 1, then y = 101 = 10. ∴ 2x = 10 x = 5 Check the answer obtained.  Checking: L.H.S. = log [2(5)] = log 10 = 1 = R.H.S.

How to solve an equation with both sides involving common logarithms? You can make use of the following property.

How to solve an equation with both sides involving common logarithms? If log x = log y, then x = y.

For example, ∵ log (x + 4) = log 9 ∴ x + 4 = 9 x = 5 If log x = log y, then x = y. ∴ x + 4 = 9 x = 5  Checking: L.H.S. = log (5 + 4) = log 9 = R.H.S.

Follow-up question Solve log (x - 1) + log 5 = 3 . ∵ ∴ log [5(x - 1)] = 3 ◄ log (MN) = log M + log N 5(x - 1) = 1000 ◄ If log y = 3, then y = 103 = 1000. x - 1 = 200 x = 201

Do you know how to solve the exponential equation 3x = 7? 7 cannot be expressed as a rational power of 3. How to solve this equation?

You can take common logarithms on both sides first. 7 cannot be expressed as a rational power of 3. How to solve this equation?

3x  7 log 3x  log 7 x log 3  log 7 3 log 7 = x I see. Let me try to solve the equation. Take common logarithms on both sides. 3x  7 log 3x  log 7 x log 3  log 7 ◄ log Mn = n log M 3 log 7 = x 77 . 1 = (cor. to 3 sig. fig.)

Follow-up question Solve 4x + 1 = 5, correct to 3 significant figures. Take common logarithms on both sides. ◄ 5 log 4 1 = + x 5 log 4 1) ( = + x ◄ log Mn = n log M 4 log 5 1 = + x 1 4 log 5 - = x 0.161 = (cor. to 3 sig. fig.)