WEBVTT
00:00:01.040 --> 00:00:11.840
Solve 20 is equal to 50 multiplied by 1.029 to the power of 𝑥 for 𝑥, giving your answer to three decimal places.
00:00:12.600 --> 00:00:17.560
In order to calculate the exponent of any exponential function, we will need to use logarithms.
00:00:18.040 --> 00:00:23.320
Before doing this in this question, we will divide both sides of the equation by 50.
00:00:23.960 --> 00:00:26.440
The left-hand side simplifies to two-fifths.
00:00:26.720 --> 00:00:30.800
As a decimal, this is equal to 0.4.
00:00:31.360 --> 00:00:35.600
On the right-hand side, the 50s cancel.
00:00:36.080 --> 00:00:40.920
So, we are left with 1.029 to the power of 𝑥.
00:00:41.560 --> 00:00:44.840
There are two ways of approaching the problem from this point.
00:00:45.400 --> 00:00:53.120
Firstly, we could recall that if 𝑎 to the power of 𝑥 is equal to 𝑏, then 𝑥 is equal to log base 𝑎 of 𝑏.
00:00:53.680 --> 00:00:59.800
In our question, 𝑎 is equal to 1.029 and 𝑏 is equal to 0.4.
00:01:00.320 --> 00:01:06.520
This means that 𝑥 is equal to log to the base 1.029 of 0.4.
00:01:07.280 --> 00:01:18.760
Typing this into the calculator gives us negative 32.052194 and so on.
00:01:19.280 --> 00:01:26.720
As we need to give our answer to three decimal places, the one is the deciding number.
00:01:27.200 --> 00:01:37.800
As this is less than five, 𝑥 is equal to negative 32.052 to three decimal places.
00:01:38.280 --> 00:01:43.640
We could check this answer by substituting our value of 𝑥 into the original equation.
00:01:44.480 --> 00:01:55.160
An alternative method from the line 0.4 is equal to 1.029 to the power of 𝑥 would be to take logs of both sides.
00:01:55.800 --> 00:02:01.000
We could then use one of our laws of logarithms to simplify the right-hand side.
00:02:01.400 --> 00:02:05.760
log 𝑎 to the power of 𝑥 is equal to 𝑥 multiplied by log 𝑎.
00:02:06.360 --> 00:02:12.800
The right-hand side simplifies to 𝑥 multiplied by log of 1.029.
00:02:13.240 --> 00:02:23.200
Dividing both sides of the equation by this gives us 𝑥 is equal to log 0.4 divided by log of 1.029.
00:02:23.600 --> 00:02:32.480
Once again, this gives us negative 32.052 to three decimal places.